Advances in numerical and applied mathematics

NASA Contraaor Report 178075

ICASE REPORT NO. 86-18 nasa-cr-i 78076

19860017080

ICASE

ADVANCES IN NUMERICAL AND APPLIED MATHEMATICS

Edited by

J. C. South, Jr.

and

M. Y. Hussaini

Contract Nos. NASl-17070 and NASl-18107

March 1986 ^ -*:: '•'

H -.'.'PTOn V;

INSTITUTE FOR COMPUTER APPLICATIONS IN SCIENCE AND ENGINEERING
NASA Langley Research Center, Hampton, Virginia 23665

Operated by the Universities Space Research Association.

fVlASA

National Aeronautics and
Space Administration

Langley Research Center

Hampton, Virginia 23665

3 1176 01306 8508

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ADVANCES IN NUMERICAL AND APPLIED MATHEMATICS

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Dr. Milton E. Rose

DEDICATION

Dr. Milton E. Rose began his mathematical career in numerical analysis at
the start of the computer era. He received the Ph.D. degree from New York
University where he studied under Richard Courant. There, using a Univac I, he
helped demonstrate the feasibility of studying floods in a large river system with
dams and power stations (Ohio, Tennessee, and Mississippi rivers). His research
has continued to emphasize the importance of developing efficient approximation
methods for the numerical treatment of partial differential equations while keep-
ing physical ideas at the forefront. His work has served as an inspiration for two
generations of colleagues. In particular, his treatment of "Stefan problems," using
enthalpy rather than temperature, has become the standard practice in the field.

Dr. Rose has continued his research while engaged in eoi active administra-
tive career. He has served as Head of the Applied Mathematics Division,
Brookhaven National Laboratory; Head of the Mathematical Sciences Section,
National Science Foundation; Head of the Office of Computing Activities,
National Science Foundation; Chairman of the Mathematics Department,
Colorado State University; Chief of the Mathematics and Geosciences Branch,
Energy Research and Development Administration; and has served as Director of
the Institute for Computer Applications in Science and Engineering (ICASE)
since September 1977. His administrative efforts produced remarkable improve-
ments in the fields of computer applications that he managed for the U. S.
government.

At ICASE, Dr. Rose has nurtured and brought to maturity an activity that
has gained international recognition for its breadth and intellectual content.

On the occasion of his 60th birthday, a few of Dr. Rose's friends have pro-
duced this volume to show their appreciation for his wise and happy guidance and
to challenge him to keep it up for the next 60.

Eugene Isaacson
March 1986

XIX

FOREWORD

This volume contains 21 research papers dedicated to Milton E. Rose on the
occasion of his 60th birthday. The contributors are mathematicians and fluid
dynamicists who have known and worked with Milt Rose during his tenure as
Director of ICASE.

These research papers cover some recent developments in numerical analysis
and computational fluid dynamics. Some of these studies are of a fundamental
nature. They address basic issues such as intermediate boundary conditions for
approximate factorization schemes, existence and uniqueness of steady states for
time-dependent problems, pitfalls of implicit time stepping, etc. The other stu-
dies deal with modern numerical methods such as total-variation-diminishing
schemes, higher order variants of vortex and particle methods, spectral multi-
domain techniques, and front-tracking techniques. There is also a paper on adap-
tive grids. The fluid dynamics papers treat the classical problems of incompressi-
ble flows in curved pipes, vortex breakdown, and transonic flows.

The editors would like to take this opportunity to thank the authors for their
excellent contributions and their promptness for meeting deadlines.

JCS and MYH
March 1986

V

TABLE OF CONTENTS

Section I

Convergence to Steady State of Solutions of Burgers' Equation

Gunilla Kreiss and Heinz- Otto Kretaa 1

Stability Analysis of Intermediate Boundary Conditions in Approximate
Factorization Schemes
Jerry C. South, Mohamed M. Hafez, and David Gottlieb 30

Multiple Steady States for Characteristic Initial Value Problems

M. D. Salas, S. Abarbanel, and D. Gottlieb 56

A Minimum Entropy Principle in the Gas Dynamics Equation

Eitan Tadmor 100

A Spectral Multi-Domain Method for the Solution of Hyperbolic Systems

David A. Kopriva 119

On Substructuring Algorithms and Solution Techniques for the Numerical
Approximation of Partial Differential Equations
M. D. Gunzburger and R. A. Nicolaides 165

Section II

Multiple Laminar Flows Through Curved Pipes

Zhong-hua Yang and H. B. Keller 196

Calculations of the Stability of Some Axisymmetric Flows Proposed as a Model
of Vortex Breakdown
Nessan Mac Giolla Mhuiris 229

Numerical Study of Vortex Breakdown

M. Hafez, G. Kuruvila, and M. D. Salas 264

Multigrid Method for a Vortex Breakdown Simulation

Shlomo Ta'asan 291

VXl

Construction of Higher Order Accurate Vortex and Particle Methods

R. A. Nicolaides 312

Pseudo-Time Algorithms for the Navier-Stokes Equations

R. C. Swanson and E. Turkel 331

Section III

Conditions for the Construction of Multi-Point Total Variation Diminishing
Difference Schemes
Antony Jameson and Peter D. Lax 361

Some Results on Uniformly High Order Accurate Essentially Non-oscillatory
Scheme

Ami Harten, Stanley Osher, Bjorn Engquist, and Sukumar R. Chakravarthy
383

On Numerical Dispersion by Upwind Differencing

Bram van Leer 437

Aztec: A Front Tracking Code Based on Godunov's Method

Blair K. Swartz and Burton Wendroff 449

Least Squares Finite Element Simulation of Transonic Flows

T. F. Chen and G. J. Fix 467

The Weak Element Method Applied to Helmholtz Type Equations

Charles I. Goldstein 495

The Local Redistribution of Points Along Curves for Numerical Grid
Generation
Peter R. Eiseman 533

On Similarity Solutions of a Boundary Layer Problem with an Upstream
Moving Wall
M. Y. Hussaini, W. D. Lakin, and A. Nachman 557

On the Advantages of the Vorticity- Velocity Formulation of the Equations of
Fluid Dynamics
Charles G. Speziale 58i

Vlll

CONVERGENCE TO STEADY STATE OF SOLUTIONS OF BURGERS' EQUATION

Gunilla Kreiss

Royal Institute of Technology

Stockholm, Sweden

and

Helnz-Otto Kreiss

California Institute of Technology

Pasadena, California

Abstract

Consider, the initial-boundary value problem for Burgers' equation. It is
shown that its solutions converge, in time, to a unique steady state. The
speed of the convergence depends on the boundary conditions and can be
exponentially slow. Methods to speed up the rate of convergence are also
discussed.

Research was partially supported by the Office of Naval Research under
N00014-83-K-0422 and National Science Foundation Grant DMS-8312264.
Additional support was provided by the National Aeronautics and Space
Administration under NASA Contract No. NASl-17070 while the authors were in
residence at the Institute for Computer Applications in Science and
Engineering, NASA Langley Research Center, Hampton, VA 23665-5225.

1. Introduction. In many gasdynamical problems one tries to calculate the steady state
solution by solving the corresponding time dependant problem. One hopes that for i — cxj the
solution converges to a unique steady state. Recently, M. D. Salas, S. Abarbanel and D. Gottlieb
[l]considered the initial-boundary value problem

«f + 2 ("^)=' = fi^)' < ^ 0. < a; < TT,

(1.1)
u{x,0) =g{x).

They used

f{x) =smxcosx, g{x) =bsinx, < 6,

and showed that the solution u{x, t) of the above problem converges to a steady state v{x), as
t -* CO, but that v{x) depends on the initial data.
In this paper we consider the viscous problem

"t + 2 ("^)^ = ^"^^ + /(^). ^ ^ 0. < a; < 1, e > 0, (1.2a)

with initial and boundary conditions

^(^,0) =gix),
u{0,t)=a, u(l,0=6,

(1.26)

(1.3)

and the corresponding steady state problem

■^{y'^)x=^eyxx+f{x), 0<x<l, e>0,

y(o) = a, y(i)=6.

For simplicity we restrict ourselves to two cases:

1) a > > 6, a > -6, f{x) = 0,

2) a = 6 = 0, f is such that there exists an a with < a < 1 such that f{x) > for < z < q,
fix) < for a < X < 1, /(O) = /(I) = 0, ^(0) > /o > and ^(1) > /o.

We will show that (1.3) has a unique solution and discuss the properties of y(z). We shall
also show that in all cases we consider, the limit of y{x) as e — *• exists. Thus, if

^limti(x,t) =y(x)

exists, we obtain a unique steady state solution of the inviscid equation (1.1) if we first let
t -^ oo and then e — ^ 0. This is in contrast to the procedure in [ l], where the two limit
procedures are taken in the reverse order.

We shall prove that the eigenvalues of the eigenvalue problem

\<p = -Mx + e<p^^, <piO) = <pil) = 0, (1.4)

are all negative. Therefore, the solution of (1.2) converges to the solution of (1.3) provided
u{x, 0) = g{x) is sufficiently close to y{x). In another paper we shall prove that u{x, t) converges
to y(x) as f — »■ oo for arbitrary initial data. The speed of convergence is determined by the
eigenvalues, Ay, of (1.4). We shall show that the eigenvalue distribution depends on f{x) and
on a, 6 in the following way:

There is a constant c > which does not depend on e such that

(1) if a>-6, / = then > -c/e > Ai > A2 > • • •

(2) if a = -6, / = then - Ai = ©(e"^/') > 0, -c/e > A2 > A3 > • • •

(1.5)

(3) if a = 6 = 0, / f{x)dx ^ 0, then - c> Ai > A2 > • • •

Jo

(4) if a = 6 = 0, / f{x)dx = 0, then - Ai = 0(e"^/^) > 0, -c> A2 > A3 > • • •

We expect a reasonable speed of convergence in the first and third case, while in the second
and fourth case the speed should be extremely slow due to the eigenvalue — Ai = 0{e~^i'). This
is confirmed by numerical experiments. We see that at first u[x, t) quite rapidly approches the
same limit as the inviscid equation (1.1), which consists of solutions of the stationary equation

connected by a shock. Once the viscous shock has been formed, the solution of (1.2) becomes
quasi-stationary and the shock creeps extremely slowly to the "right" position. We can ex-
plain the behavior, because by linearizing around the quasistationary solution we find that the
eigenvalues of the corresponding eigenvalue problem have a similar distribution as earlier.

If — Ai = 0(e~^/^) then the speed of convergence is so slow that the above method to
calculate the steady state is impractical, see figures (1) and (3). However, we can use the same
technique as Hafez, Parlette and Salas in [2] to speed up the convergence. See figures (2) and

Unfortunatly, not only the speed of convergence but also the condition number of the
stationary problem deteriorates. We have to calculate with 0{e}l') decimals to obtain correct

results. To avoid an excessive number of decimals we have used a quite large e in our numerical
calculations.

The situation becomes much better in a two dimensional case, which we discuss in the last
section. Now there is a whole sequence of eigenvalues

-/,.-=nr-,-2

/^i;=o(re), y = i,2

close to zero. However, they are only algebraically and not exponentially close to zero. We
indicate how to modify the procedure to accellerate the speed of convergence.

We beheve that the viscous model (1.2) better explains what happens in actual calculations
than the inviscid equation (1.1). Practically all numerical methods have some viscosity built in.
Also, from a physical point of view, the solution we are interested in is the limit of solutions of
a viscous equation.

Finally we want to point out that the appearance of small eigenvalues has also been
observed by D. Brown, W. Kath, H. O. Kreiss and W. Henshaw, M. Naughton (private com-
munication).

2. Uniqueness,existence and properties of the steady state solution. We start
with uniqueness, which can be proven by standard techniques.

Lemma 2.1. If the steady equation (1.3) has a solution, then it is unique.
Proof. Let u, v be two solutions. Then w = u — v is the solution of

-(pw)i = ewii, p = u + v, w{0)=w{l)-0. (2.1)

]i w ^0 then the zeros of «; are isolated. Let x with < x < 1 be the flrst zero to the right
of X = 0. Without restriction we can assume that ly > for < a: < 2, i.e. ^^(O) > and
Wx{x) < 0. Integration of (2.1) gives us

-e{\^x{x)\ + \w,iO)\) = eKJS = i[p«;]g = 0.

Thus u;i(0) = Wxix) = 0. We can consider (2.1) as an initial value problem with initial data
u;(0) = Wx{0) = whose solution is w{x) = 0, and the lemma is proved.

We shall now discuss the properties of the solution. Let us start with the case f{x) = 0,
a > > 6, a > —b. Integrating (1.3) gives us

^yx = -y^-c, o<x<i,

(2.2)
y(o) = a.

The constant c has to be determined so that y(l) = 6. We necessarily have c = cf2/2 > 0^/2,
because with c < a^/2, t/i > for all x, and y(l) = fc cannot be satisfied. We can solve
equation (2.2) explicitly. This is done by writing (2.2) in the form

yW

'^lwh=i'''

I.e.

a + d y(x)-d ^ j^/^
^a-d'^y{x)^d'

Therefore y(l) = 6 implies d = a + 0(e~*/'), and

1 — re~''(*~^)/* a — 6

yix) = a 7T — 77-, with r = — —r. (2.3)

Away from the boundary layer at x=l we have y{x) = a + 0(e~'^^^~^^f'). Thus, for e — ► 0, y(x)
converges to a for < s < 1.

If a = —6 we consider (2.2) on the interval < i < i, with boundary conditions y(0) =
fl) 2/(2) — ^ ^^^ obtain a solution yi{x) of the form (2.3). The solution on the whole interval
is given by

^^ ^~l-yi(l-a:), ifi<x<l.

In figures (9) and (10) we have plotted y(i) for two different sets of boundary values.

Consider case 2, where / only vanishes at i = 0, a, 1 and a = 6 = 0. Without restrictions
we can assume that

1
//(x)>0. (2.4)

If this is not true, we transform the problem by introducing new variables,

5 = 1 - X, / = -/, y = -y.

The new problem satisfies (2.4).

Lemma 2.2. Let y[x) be the solution of (1.3), F{x) = f^ fiO^^ and H^) = \/2F(x).
Then

yx(l) < yx{0) <Ku K,= max {lh^(x)|} + l/i.(0)|.

Proof. Integration of (1.3) gives

e(yz-yz(0)) = iy2-F,

(2.5)

y(o)=o,

where t/i(0) is determined by y(l) =0. If u = y - /i, then u is the solution of

Ux = 1/2(0) -hx + e~^uh + -e~'«^,
«(0) = 0.

Assume that ya;(0) > /fi. It follows that ya;(0) -/ij(a;) is positive and thus u and Ua; are positive
for all a; > 0. In particular u(l) > and y(l) = ii(l) + h{\) > 0, which contradicts y(l) = 0.
Thusy^(O) <Ki. Also

^yxil)=ey^{0)-F{l)<ey40).
This proves the lemma.

Lemma 2.3. Let y{x) be the solution of (L3) and let e be sufficiently small. If F{1) >
then y{x) > for < a; < 1 and y{x) has exactly one maximum. If F(l) = then there exists
an X with < x < 1 such that y{x) > for < a; < x, and y(x) < for x < a: < 1. Also y(x)
has exactly one minimum and one inaximum. In both cases \y{x)\ < max|F(a;)l.

Proof. At extrema y^ = and

{< for < a; < a,
= for a; = a, (2.6)

> for a < a; < L

Thus y cannot have a minimum to the left of a maximum. Since y(0) = y(l) = there are only
three possibilities, namely

y>0 for 0<x<l, y has exactly one maximum, (2.7a)

y < for < a; < 1, y has exactly one minimum, (2.76)

y > for 0<x<x, < x < 1,

y<0 for X <x <1, y has exactly one maximum and one minimum.

(2.7c)

We shall prove that if F(l) > then (b) and (c) are not possible, and that if F(l) =
then (a) and (b) are not possible.

Let F(l) > 0. Suppose (2.7b) holds. Then

yx{0)<0, y:r(l)>0.

By (2.5)

0<£(y.(l)-yx(0)) = -F(l)<0. (2.8)

This is a contradiction, so (2.7b) cannot hold. Now supposse (2.7c) is valid. Then yx{0) >

and by (2.8)

y.(0)>e-iF(l).

If e is small enough this is impossible by lemma 2.2.

Let F(l) = 0. Assume that (2.7a) or (2.7b) are valid. By (2.8) y^iO) = yx{l), which is
only possible if yx{0) = yx{l) =0. Differentiating (1.3) gives us

eyxxx = yyxx + [yxf - fx- (2-9)

Thus

y(0) = yx{0) = yxx{0) = 0, y^xxiO) < 0,
y(l) = yx{l) = yxxil) = 0, yxxxil) < 0.

This implies that y must change sign at least once, which contradicts the assumption, and
therefore (2.7c) must hold.

It remains to show that \y{x)\ is bounded by max|F(x)|. Since y(0) = y(l) = 0, the
maximum absolute value of y is found at a local extrema, where y^ = 0. Thus, from (2.5) it
follows that

\y{x)\ < mj^^ \F{x) - eyM\ < mj^^ \F{^)\-

This finishes the proof.

We can use the usual singular perturbation methods to discuss the behavior of the solution
in detail, see for ex. [3].

Theorem 2.1. Let F(l) > 0, assume that (1.3) has a solution and that e is sufficiently
small. Then y(x) has a boundary layer at x = 1. For 1 - 0(e| log(e)|) < x < 1, y(x) is close to
w{x) which is the solution of

ewx = ^w'^-Fil), -oo<x<l, wil)=0. (2.9)

In any interval 0<a;o<x<l— 0(e| log(e)|)

y{x) = hix) + eui (x, e), h{x) = sj2F{x) =: xg(x), (2.10)

where ui and its derivatives are bounded independantly of e. For < x < xq < a we have

y(x) = h{x)+eu{x), x = xfy/i, (2,ii)

where u and the derivatives d^u/dx" are bounded independantly of e. Thus, for e — 0, y{x)
converges to h[x) for < x < 1.

Proof. We indicate only the proof of (2.11). In the proof we shaU use hj2 and / to
denote the inteirals < x < 1, 1 < x < xo/V? and < x < Xo/y/i, respectively. We shaU also

use

/:=max|/(x)|,
It/

where / is an interval.

We introduce a new variable in (1.3),

y(x) =/i(x) + etx(x/v^.
This gives us

Uii-{3:g{x)-\-^u)ui-h^u = -h^^, < x < xo/Ve, «(0) = 0, «(xo/v^ = «o, (2.12)

where uq = ui(xo,e) is bounded independantly of e. Prom xo < a and the assumtion /r(0) >
/o > it foUows that h^{x) > /lo > for < x < xq. Therefore we can use the maximum
principle. The maximum of u is found either on the boundary or at a local extrema, where
Ui = 0. At local extrema

W\ < 1^1 < ^\\h,4^)\\j =: a.

Thus

||w||/<max(uo,Qr). (2.13)

Next we want to estimate \\ui\\r. First we consider the interval /j = [0, 1]. By (2.12) and
(2.13) there are constants Ci and C2 such that

IK5II/, <C'i||u5||;, +C2.

It is well known, see Landau [4], that one can estimate ||ui||/, in terms of ||u||/,, and ||u5j||/,,
i.e. for every constant 6 there is a constant C{S) such that

IKIk <<5||ui5||/.+C(5)IH|;,.

Thus for 6 = ^(Ci)-' we obtain a bound for ||u55||/,, which gives us a bound for ||u5||/,.
Especially, |u2(l)| is bounded.

In the remaining interval I2 = [l,xo/y/£], we have

F>F{Vi)=eM0)il + O{^)).
Thus

i.e. for sufficiently small y/i

At local extrema of uj, ujj = and we have, by (2.12),

Thus

||u2||/.<max(|«5(l)|,|u5(^)U),

and Ux is bounded independantly of e in the whole interval. By differentiating (2.12) bounds
for higher derivatives of u can be obtained.

It is also clear that as e — ► 0, y[x) converges to h{x). This finishes the proof.

If F(\) = then the solution switches at x from \/2F + 0{e) to -\/2F + 0(e). In each
subinterval < a; < x and x <x <1 the local behavior of the solution is of the same type as
in the first case. As e — ^ 0, y{x) converges to h{x) for < a; < x and to —h{x) foTx<x< 1.
In general, the position of x can only be obtained by detailed calculation. However, if /(x) is
antisymmetric around x = | then x = i. This is the only case we consider.

We shall now discuss the existence of a solution. For this we need two lemmata.

Lemma 2.4. For sufficiently large e the steady state equation (1.2) has a solution.

Proof. By integrating (1.3) twice, we can write the equation in the form

X X

y(^) = ^rij y^iOd^ -vj F{Od^ + T)xco, V = 1/^

1 1

IfyHOd^-jFiOd^ + co^O,

or after the change of variable y = rjy

X X

y{^) = y^fyHOdi-jFiOd^ + xco,

1 1

\n^ j fmi- j F{Odi + CO =0.

For r? = the above equations liave a unique solution. Therefore the same is true for all
sufficiently small rj. This proves the lemma.

Lemma 2.5. Let p{x) be a smooth function. Consider the eigenvalue problem

X<P = -{p(p)x + e<pxx, cp{0) = (p{l) = 0. (2.14)

The eigenvalues are real and negative.

Proof. We introduce a new variable ipix) by

V?(x) = e 1 rjj{x),

and obtain

XxP = ei/.^^ - crp =: Lxp, c{x) = -px{x) + — (p(x))^,
V.(0) =,/>(!) =0.

(2.15)

(2.15) is selfadjoint and therefore the eigenvalues are real. Let <p ^^ 0, A be a solution of (2.14),
and let x be the first zero of (p to the right of a; = 0. We can assume that ^ > for < a; < 5.
Thus (px{0) > and (px{x) < 0, and integration of (2.14) gives ys

X

A / <p{x)dx = e[ipx]o < 0.

It follows that A < 0. K A = 0, the only possible solution of (2.14) would be (p{x) = 0. Thus
A < 0, which proves the lemma.

Now we can prove

Theorem 2.2. The equation (1.3) has a unique solution for all e > 0.

Proof. We have already shown that (1.3) has a solution for sufficiently large e. We will
now employ continuation in e to prove existance for all e > 0. Assume we have shown existance
for £ > e. We want to show that there is a solution for e = e. By lemma 2.3 the solutions of
(1.3) are unifonnly bounded for e < e < e + 1. Therefore the same is true for the first three
derivatives. Thus we can select a sequence of solutions

y{x,e^), i/ = l,2,..., lim£i,=e,

such that

,l™od^2/(^'^-) = d^2/(=^'^)' i = 0,l,2

10

and y{x,e) is the desired solution.Linearizing the equation around y{x,e) gives us
{y{x,e)6y)x = e{6y)xx + (e - e)y(a;,e), Sy{0) = 6y{l) = 0.

By the previous lemma A = is not an eigenvalue of the above equation and therefore we can
solve (1.3) for all sufficiently small e — e. This proves the theorem.

3. Speed of convergence. In this section we want to discuss the speed of convergence
to steady state. We assume that the initial data g{x) of (1.3) are sufficiently close to the
solution of the steady problem, so that we only have to discuss the behavior of the solutions of
the linearized equation

wt + {yw)x = ewxx, < a; < 1, t>0,

w{x,0)=g{x), (3.1)

w{0,t)=w{l,t)=0.

To determine the speed of convergence we study the distribution of eigenvalues of

\<p + {y<p)x = e<pxx, <p{0) = <p(l) = 0. (3.2)

Theorem 3.1 . The eigenvalues of (3.2) are real and negative and their distribution is
given by (1.5).

Proof • Lemma 2.5 tells us that the eigenvalues are real and negative. First we consider
the case / = 0, a > —b. We write (3.2) in the selfadjoint form (2.15) with p = y. Let A = Ai
be the largest eigenvalue. The corresponding eigenfunction xpi does not change sign, and we
can assume that tpi > for < a; < 1 and that max |i/'i(a:)| = 1. We assume that Ai > — a^/8e.
Then there is a constant K such that c(a;) + Ai > for > s > 1 — Ke. Thus rpi is monotone
in the interval < a; < 1 — Ke, and therefore xpi must have its maximum in the remaining
interval, 1 — Ke <x<l. By assumption maxV'i(a;) = 1 and therefore there must be a constant
5 > such that tpix{l) < —6/e. Now consider the corresponding eigenfunction

<pi (x) = e 1 Vi (x), <pix{l) = tAii(l), < <pi (x) < Vi (x).

Integrating (3.2) gives us

-6 > e{(pix{l) - <pix{0)) = Ai / (pidx >

Jo

>Ai r e^'~' fi ^'^Ux = Xisd.
Jo

11

Thus

and the theorem is proven for this case.

When / ^ 0, a = 6 = 0, ;ind /q f{x)dx > the corresponding estimate follows in the same
way, since by theorem 2. 1 there are constants Co > and K such that

c{x) = -{h:,{x)+0{y/e) + -e-^{h{x)+0{^ye)f)>Co>0 for 0<x<l-Ke.

We now consider the antisymmetric case when a = —6, / = 0ora = 6 = and f{x) is
antisymmetric around a; = 2- We want to show that

-Ai = 0(e-ie-*/').

We shall use the fact that for our selfadjoint eigenvalue problem (2.15) the eigenvalue with the
smallest absolute value, Ai, satisfies

"■'- MiT'

for any smooth function <t>^0 satisfying the boundary conditions. We chose

(}){x) = e '/2 _ g

as trial function. y{x) is antisymmetric around x = ^, and <f){0) = (f>{l) = 0. Also

Both 4? and (^e ^y^ + \^yx)^ aje symmetric around a; = |. Therefore

= 2/e--'/o"^^''^(eK'/o'^^^-l)2d:.,

12

and by (2.5) and theorem 2.1

A? < 1'^" = 2 < ^2.-2 --2D/r

1 — IUII2 i 5: i^ e e ' ,

/ (e _ 1)2^3.

where C > , Z) > are constants which do not depend on e.

We shall now estimate the size of the second eigenvalue for the case with an interior
boundary layer at x = i. By assumption y(x) is antisymmetric around x = ^. Consider the
eigenvalue problem (3.2) on half the interval, < a; < i, and denote its solutions by

<Pi{x), A,-, t = l,2....

We know that (pi has t - 1 sign changes, and we have already shown how the Aj's are bounded
away from zero. The function

, . i <Pi{x) forO<a;< i . , ^

<P2i{x) = <^ : 1. , iZ Z-i^ 1 = 1,2...,

[ -(pi(x - i) for i < a; < 1 ' ' '

will satisfy (3.2) on the fuU interval, < x < 1 with A = A2i = Aj. Also (p2i changes sign
2(t - 1) + 1 times. Thus (p2i is the 2i*'» eigenfunction and A2,- is the 2t*'' eigenvalue. Therefore
A2 is bounded away from zero. This finishes the proof.

4. Numerical results. We shall discuss difference approximations for the time depen-
dant problem (1.2) and the eigenvalue problem (3.2). We introduce gridpoints

{xi = ih,tj=3k), t=0,l,... y = o,i N, ^ = 1

where iV is a natural number and A; > is the time step. We also introduce gridfunctions

u^ =u{xi,tj).
We approximate (1.2) by the usual implicit method

(/ - efcZ)+D_)ui+i + \kDo{J^'f = t^' + kfi, i = l,2,...,N-l (4.1)

13

with initial and boundary conditions

«? = ff.-, t = l,2,...,^'-l,

Here

h^D+D-Ui = Ui+t—2ui + Ui-i and 2/iDo(«i)^ = («.+i)^ - («.-i)^

denote the usual centered difference operators. At every time step one has to solve a nonlinear
system to determine u^"*"^ . This is done by the iteration

iI-ekD+D-)uf+'^=~kDoiuf^)'' + v^i+kfi, 1 = 0,1... , (4.2)

where «(°) is choosen by a predictor process.

In all our experiments the solution of (4.1) converges to a steady state solution. However,
the speed of convergence depends on the location of the shock. If the shock is located at the
boundary, corresponding to the first and third case of (1.5), then the convergence to steady
state is quite rapid. See figure (5). If on the other hand the shock is located in the interior,
corresponding to the other cases of (1.5), the convergence is, in general, very slow. When the
shock is formed at an early stage it is in general in the "wrong" place, depending on the initial
data. From then on, the the shock moves slowly to the correct position. See figures (1),(3). This
process can be considered quasi-stationary, which makes it possible to use the same convergence
acceleration as in [2].

Formally we can write our iteration (4.1) as

i?(«"+^) = u"+i - u" := r". (4.3)

We can linearize the realation and obtain

(/ - L)r"+^ = r". (4.4)

In our case

Lri = ekD+D-ri - A;Do(«;'+V,). (4.5)

This is a discretization of the right hand side of the eigenvalue problem (2.14), with p = u". If
the process is quasi-stationary we can consider L to be independant of n. Then we have

r"+> = (/-L)--'r"

and

p-i
u"+P = tx" + X)(^-^)"-''-"-

If the eigenvalues A,-, of L are negative the eigenvalues /c^, of (/ — L)~^ satisfy |/c,| < 1 and

lim u"+'' = «" + (/-(/- L)-^)-'r" = «" -f (/ - L-')r'' (4.6)

p— »oo

14

Instead of taking a large number of time steps we caji take one large step, which we call an
extrapolation step. We put

« = u" + /9e, (4.7)

where e is the solution of the equation

Le=(L-/)r", (4.8)

and /9 is a stabilizing parameter. We choose p in such a way that H{u'*-\-pe) has no component
in the direction of e, i.e.

<i/(«"+^e),e>=0,

where (•, •) denotes the usual inner product. There are other possible choices, for example
choose /? such that

||H(u"+^e)|| = imn||ff(u"+^e)||.
p

Of course (4.7) is not the steady solution we are seeking. We use the new u to restart the time
iteration, and make a new extrapolation step once a new quasi-stationary state is reached. In
our experiments we use an a priori fixed number of time steps between the extrapolation steps.
Better strategies are under development.

We have calculated the first eigenvalues and eigenvectors of the discrete linearized operator
(4.5), provided u^^^ is the discrete steady state solution. The calculations show that the
eigenvalues are negative and their distribution is of the same type as for the corresponding
continous case. See table (1). In figures (6), (7) the first few eigenvectore are plotted. Note that
in the case of an interior shock the first eigenvector is exponentially small away from the shock
region. Also, we have no doubt, and it is confirmed by the calculations, that the position of the
shock does not change the nature of the eigenvalue distribution. In fact, in the proof of theorem
3.1, y can be replaced by any function of the sjmie structure.

In our case, when the shock is located in the interior, (/ - L)~* has only one eigenvalue,
Ki, close to zero. All other eigenvalues are small. Therefore, when we have reached the quasi-
stationary state, r" is in the direction of the eigenvector corresponding to /ci. See figure (8).
Therefore we do not need to solve (4.8), and instead of (4.7) we use

tx = «" + /9r". (4.9)

In figures (2), (4) we have plotted u at different time stages to show how the convergence is
accelerated.

15

5. A twodltnensional case. Consider the following problem

«t + (2"^)* = e(«~ + «yy). 0<a<l, 0<y<l, f>0,

u(0,y,t)=a, u{l,y,t) = -a, a > 0, (5.1)

u(x,0, t) — u{x,l, t) = w{x),
u{x,y,0) = g{x,y),

where W{x) is the solution of the one dimensional problem (1.3) with 6 = —a, and f{x) = 0.
See (2.3). A steady solution of (5.1) is u{x,y) = w{x).

The speed of convergence can be studied by analyzing the corresponding eigenvalue prob-
lem

ficp + (w<p)x = £(<Pxx + <Pyy), <)? = on the boundary. (5.2)

We caji solve (5.2) by separation of variables. Let <p{x,y) = X{x)Y{y). Then

(wXy - eX" = XX, X(0) = X(l) = 0, (5.3a)

Y" = -qY, r(0)=r(l) = 0, (5.36)

with n = \~eq. We recognize (5.3a) as (3.2). Therefore -Ai = 0(e-^l') and -Ay > 0{l/e),
j = 2,3, — We can solve (5.3b). The solution is

Yj{y) = sm(JTry), q, = [jirf, j = 1, 2 . . . .

There is a whole sequence of eigenvalues, /ny, of order 0(e). The eigenfunctions corresponding
to this sequence, (pij, will be exponentially small away from the shock. All other eigenvalues
will be of order 0(l/e).

We expect that the time iteration will again lead to a quasi-stationary state, and that
the residual will be composed of eigenfunctions corresponding to the eigenvalues of order 0(e).
Therefore e in (4.8) will be of the same form, and we can replace all components of e away from
the shock by zero, thus obtaining a linear system of equations of order N instead of N^. More
details will be given in another paper.

16

REFERENCES
[1] M. D. Salas, S. Abarbanel, D. Gottlieb, Multiple steady states for characteristic initial value

problems, lease report No 84-57 , NASA CR-172486, November 1984.
[2] M. Hafez, E. Parlette, M. Salas, Convergence acceleration of iterative solutions for transonic

flow computations, AIAA 85-1641.
(3) J. D. Cole, J. Kevorkian, Perturbation methods in Applied Mathematics, Springer 1981.
[4] E. Landau, Einige Ungleichungen fur zweimal differenzierbare Funktionen, Proc, London

Math. Soc. 13(1913) 43-49.

17

Table 1.

Eigenvalues of the eigenvzlueproblem (3.2), y is the solution of (1.3). Three different cases were treated.
The discretization is done according to (4.5), with N = 100 gridpoints. The eigenvalues were found using
inverse iteration. Eigenvectors corresponding to case (l) are plotted in figure (6a,b).

Ai

A2

^3

f{x) = sm(27rx)/2
a = b =
t = 0.04

-8.64-10-3

-4.34

-5.32

f{x) = sm{2nx)/2
a = b =
e = 0.02

-4.62 • 10-^

-5.617

-5.622

fix) =

a = l, 6 = -l

e = 0.02

-1.24 -10-^

-12.8

-13.5

18

Figure 1. Convergence in time without convergence acceleration. Numerical solutions at
different time stages for the case e = 0.05, / = 0, a = 1, 6 = -1, u(i,0) = 1 + 2(e-2» - l)/(l - g-^).
Between each curve there are 200 time steps = 40 time units. The calculation is made with time step k
= 0.2 and N=50 grid points.

19

.25

5

.75

1.0

Figure 2. Convergence in time with convergence acceleration. Numerical solutions at different
time stages for the same case as in figure 1. Between each curve there are 15 time steps and one
extrapolation step. The same time step, k=0.2, and number of grid points , N=50, are used.

20

Figure 3. Convergence in time without convergence acceleration. Numerical solutions at
diflFerent time stages for the case e = 0.04, / = f sin(7ra;)cos(7ra;), a = 6 = 0, u{x,0) = isin(7rx).
Between each curve there are 100 time steps , The calculation is made with time step k = 0.1 and N=50
grid points.

21

Figure 4. Convergence in time with convergence acceleration. Numerical solutions at different
time stages for the same case as in figure 3. Between each curve there are 20 time steps and one
extrapolation step. The same time step, k=0.1, and number of grid points , N=50, are used.

22

Figure 5. Convergence when the shock is located at the boundary. Here e = 0.04, f{x) =
f sin(7rx), u{x, 0) = | sin{iTx), N = 50,k= 0.1. Between each curve there are 5 time steps.

23

Figure 6a. Eigenvectors. The first two eigenfunctions of problem (3.2), when y, the solution of (1.3),
has a shock in the interior. In this case e = 0.04, f{x) = ^ sin(7rx) cos(7rz), a = 6 = 0, A'^ = 100.

2A

.275-

-. 275 -

-.55

.25

.50

.75

1.0

Figure 6b. Eigenvectors. The third and fourth eigenfunctions of problem (3.2), when y, the solution
of (1.3), has a shock in the interior. In this case e = 0.04, f[x) = j sin(7ri) cos(7ra;), ' a = 6 = 0, N =
100.

25

Figure 7. Eigenvectors. The first two eigenvectors, ^pi and ip2, of problem (3.2), 'when y, the solution
of (1.3), has a shock a; = 1. In this case t = 0.08, J{x) = f sin(;r2;), a = 6 = 0, N= 100.

26

-. 0125 -

-.025 -

- 0375 -

Figure 8. Differences between consecutive solutions at different time stages, when s = 0.04,
/ = ^sin(7ra;)cos(7rx), a = 6 = 0, u{x,0) = |sin(7rx). Between each curve there are 100 time steps .
The calculation is made with time step k = 0.1 and N=50 grid points.

27

1.0

.5

.5

Figure 9. The solution of (1.2) when / = 0, a = l, 6 = and s = 0.05.

28

.5

1.0

Figure 10, The solution of (1.2) when / = 0, a = 1, 6 = -1 and e = 0.05.

29

STABILITY ANALYSIS OF INTERMEDIATE BOUNDARY CONDITIONS
IN APPROXIMATE FACTORIZATION SCHEMES

Jerry C. South, Jr.
NASA Langley Research Center

Mohamed M. Hafez
University of California, Davis

David Gottlieb
Brown University

Abstract

The paper discusses the role of the intermediate boundary condition in
the AF2 scheme used by Hoist for simulation of the transonic full potential
equation. We show that the treatment suggested by Hoist led to a restriction
on the time step and suggest ways to overcome this restriction. The
discussion is based on the theory developed by Gustafsson, Kreiss, and
Sundstrom and also on the von Neumann method.

Research for the third author was supported in part by the National
Aeronautics and Space Administration under NASA Contract Nos. NASl-17070 and
NASl-18107 and under AFOSR 85-0303 while he was in residence at the Institute
for Computer Applications in Science and Engineering, NASA Langley Research
Center, Hampton, VA 23665-5225.

30

INTRODUCTION

Approximate factorization schemes are widely used to obtain efficient
solutions to problems in Computational Fluid Dynamics. In many cases,
they have provided a significant increase in efficiency over previously-used
solution methods in particular problems. Some outstanding examples are the
classical Alternating-Direction-Implicit method of Peaceman and Rachford [1],
the Briley-McDonald Linearized Block Implicit scheme [2], and the Beam and
Warming [3] Approximate Factorization (AF) scheme for the compressible Navier-
Stokes equations. In the transonic potential-flow area, some AF schemes which
have significantly improved solution efficiency are the work of Ballhaus and
Steger [4], Ballhaus et al. [5], Hoist [6], [7], and Jameson [8].

All of these schemes have the common feature that the solution procedure
is broken down into a sequence of easily-implemented stages; i.e., easily-
inverted matrix factors. Each of the stages usually requires boundary
conditions for an "intermediate" variable (vector) which is not always a
consistent approximation to the solution function desired. This feature can
make satisfaction of implicit boundary conditions difficult, at best, and
impossible, at worst. Dwoyer and Thames [9] demonstrated serious boundary-
condition problems associated with the class of AF schemes called "Locally
One-Diraensional," even in explicit schemes.

The present paper further highlights the importance of intermediate
boundary conditions by focusing on a specific example — a boundary-induced
stability restriction in Hoist's AF2 scheme [6] for the transonic full-
potential equation. An analysis of the effect of the intermediate boundary
condition is given by use of the usual von Neumann method and also the methods
of Gustaffson, Kreiss, and Sundstrom [10] and Osher [11].

31

ANALYSIS

Hoist's scheme is a variation of the AF2 schemes presented in References
4 and 5. It will be referred to herein as "AF2Y," since in its implementation
the y-operator is split, rather than splitting the x-operator as in References
4 and 5. For the purpose of analyzing the intermediate boundary-condition
problem, it is illuminating to study the application of AF2Y to the two-
dimensional (2-D) Laplace's equation in a rectangle. The present analysis is
valid only for the subsonic flow condition, which is simpler by far than the
transonic case. However, it is reasonable to assume that if boundary-induced
instability is present in the subsonic case, it will also occur in the
transonic case. In practice this was true.

The Discrete Problem

The following thin-airfoil problem is thus considered: We wish to solve
the Laplace difference equation for the disturbance velocity potential

L<})., = (a6 + b6 )(}>., = (1)

jk XX yy jk ^ '

where a and b are constant coefficients and 6 and 6 are central

XX yy

difference operators; e.g.,

^xxV^ Vl,k- ^V^^j-l.k-

(2)

The boundary conditions are set on a rectangular region with Dirichlet
conditions, ((> = 0, set on three sides (left, top, and right), representing
vanishing disturbances, and a Neumann condition at the bottom boundary,

32

representing a thin-airfoil f low-tangency condition:

(t) = s(x) at y = 0. (3)

A discrete analog of Eq. (3) at k = I can be written as;

(t^^ + tU. , = 2Ays(x) (A)

where we use the following notation for one-sided, two-point differences:

y jk ^J,k+1 ^jk

Vjk "^-k - *j,k-r

(6)

The difference operator (1) requires evaluation of 6 <{) , . at the boundary
k = 1. Since this operator can be written as:

6 = ^ - ^ , (7)

yy y y

Equation (4) is used to eliminate o ^. ., which calls for a value of <^.,
below the boundary k = I. Thus, the difference operator at k = 1 is:

h„i, . . = (a6 + 2b? )6. , - 2bAys(x),
B^j,l XX y J,l

(8)

33

The AF2Y Scheme

The AF2Y scheme models a hyperbolic equation, a<^ = V (fi, and is used as
an iteration scheme:

(a + bjjy)(- ah^t^ - a6^^)A(l)^j^ = aa)L(|,^j^ (9)

where n is the iteration counter,

b^b^ = b (10)

and A(|) is the correction

The scheme is implemented in two stages:

(oc + b^^y)fjj^ = cx(oL<|,^j^ (12)

(- ah^t^ - a6^^)A<^.^ = f .^. (13)

The intermediate variable f is defined by Eq. (13). The parameter a
corresponds to a reciprocal "time" step, At" , and is usually cycled between
small and large values to obtain rapid convergence. The parameter u
corresponds roughly to a relaxation factor which is usually close to 2.

The first stage (12) is bidiagonal, proceeding from the bottom boundary,
k = 1, to the last interior row of mesh points, k = K - 1, for every j. The

34

second stage (13) is a tridiagonal solution which proceeds row-by-row, from
k = K-l to k=l, to obtain the correction A(|)., . The second stage is
initiated with the condition A*. „ = 0, corresponding to the vanishing
disturbance, ij) = 0, at k = K.

The Intermediate Boundary Condition

The main problem in implementing the scheme is how to initiate the
bidiagonal solution for f at k = 1. It seems reasonable, at first sight,
to use a derivative condition on f at the boundary, as Hoist [6] did; i.e.,

t f . , = 0. (14)

y J,l

Comparison of Eqs. (14) and (12) implies that

fj,i=-Vj,r (i^>

If this procedure is used with no further modification, it is unstable for

small values of a (or large "time" steps) and fixed lo as described next.

Stability Analysis

A von Neumann (VN) analysis shows that the interior scheme (9) is stable
for all modes under the restrictions

< (u < 2 (16)

a > 0. (17)

35

However, the boundary scheme, implied by Eqs. (15) and (13) taken together, is
another matter.

A boundary condition more general than Eq. (14) for f can be
considered. Let a "dummy-point" value for f be given as:

'i.o'-"ur (w

Then the equation for f • j^ is, from Eq. (12),

^«-'^)^j,i = «'^Vj,i-^^hfj,i (19)

and Eq. (13) yields:

f . , = (- ab_J - a6 )t,<^. , = nL^<b^ , (20)

J,l 2 y XX ^j,l B^j,l

where

n = -rnrri — r • (21)

a+bjd-y)

To carry out a VN analysis, we substitute into Eq. (20) trial solutions

<|,^ ^ = g'" gi(jpAx+kqAy) ^^2)

where i = / -1, p and q are wave numbers, and G is the amplification
factor, to obtain:

(aB + 2Ab^ - iaE)(G - 1) = -' 2fibj(A + B - iE) (23)

36

where

A=a(l-cos5)>0

B = b (1 - cos n) >

E = b sin n \ (24)

5 = pAx

n = qAy

The stability condition, |g| < 1, reduces to:

n{(A + B)[(2 - n)bjA + (a - nbj)B] + (a - nb^E^} > 0. (25)

To maintain the Inequality (25) the following stability restrictions are
easily deduced:

< n < 2 (26)

a > bj^ fi. (27)

For the case y ~ 1 » corresponding to the backward-Neumann condition
on f (Eq. (14)), restrictions (26) and (27) reduce to Eq. (16) and

a > b^ 0) (y = 1). (28)

The restriction (27) enforces a "time" step limitation on the scheme for fixed
fi, which will slow convergence; or a reduction in Jl, according to:

n < mln /2, ^\ (29)

(-^)

37

which in fact yields fast convergence and ensures stability.

It is noted that another useful type of boundary condition for f is
given by

_ a6

(30)

which gives the same form as Eq. (20) for f^ i with

, a(a) + 6) , .

" a + b^ • ^31^

Both classes of schemes are implemented by initiating the bidiagonal march for
f using Eq. (20), under restriction (29).

The restriction (29) was verified numerically in both a constant-
coefficient, Cartesian-coordinate computer code for Laplace's equation and in
the "TAIR" code [12] by using fixed values for a (i.e., no ct-cycling) and fi,
and for various values of the coefficient bp In all cases, convergence was
obtained when the restriction (29) was obeyed; and divergence occurred when it
was violated.

The experiments with the TAIR code were especially interesting, since the
coefficient by varies along the airfoil surface. The test case chosen was
the default "0"-type mesh for an NACA 0012 airfoil. It was found that the
arithmetic mean of bj^ along the surface presented the crucial condition,
rather than the maximum value, as might be expected.

The question arises as to why the TAIR code, which implements the AF2Y
scheme with the boundary condition (14), operates so well since a is cycled
between small values, which violate the restriction (28) and large values.
The answer seems to be that a is increased within several rows adjacent to

38

the boundary to a value which (in the default mesh) meets the restriction
(28), when smaller values of a are used in the remaining interior field.
This "fix" was developed empirically by the authors of Reference 12; without
this fix the code diverges. This procedure is not recommended in general,
since it requires a discontinuous change in a. The assumption in the
development of the factored scheme (9) is that a is constant throughout the
mesh.

A seemingly attractive scheme, involving a discontinuity in a at the
boundary, is as follows: Initiate the solution for f using Eq. (15) with
(0=1, and change the second stage (13) at the boundary to:

r-2bl - a5 1 Ad). , = f . , = L^ <}). ,. (32)

*■ y xx'' ^j,l j,l B '*'j,l

This procedure exactly annihilates the boundary residual (in the linear case)
and represents a fully implicit satisfaction of the surface boundary
condition. However, the factored operator at line k = 2 is no longer the
interior-point operator, since the term -ab„6 in the inner factor is
changed to -2b5 discontinuously. It is possible to analyze such a scheme
by the methods presented herein, but the line k = 2 must be considered as
part of the boundary scheme. No details will be given here, but the analysis
shows that setting oi < 4/3 at k = 2 guarantees linear stability of the
overall scheme. However, the amplification factor modulus |g| exceeds unity
only in a narrow frequency range of small n (Eq. (24)) when to > 4/3.
Numerical experiments showed no sensitivity to the value of w at k = 2.
This scheme was always stable in tests with a constant-coefficient Cartesian-
mesh code, even with oj = 1.8 at k = 2. If the scheme was unstable for

39

highly stretched grids, setting to < 4/3 at k = 2 did not stabilize
the scheme. In the variable-coefficient, nonlinear case, such a scheme is no
faster than, and not as robust as, the scheme (20) with restriction (29).

Review of the Stability Theory

It is well known that in general the von Neumann analysis at a single
line is neither sufficient nor necessary for checking stability. Trapp and
Ramshaw [13] pointed out the usefulness of the VN analysis to study boundary
schemes but recognized that no theoretical justification was known.

We wish to review briefly the stability theory for finite-difference
approximations to initial boundary-value problems. A necessary condition for
the stability of such a scheme is the Ryabenkii-Godunov condition. It states
that the numerical scheme is unstable if there exists a solution of the type

^",k = ^"'^j,k' 1^1 >1 (33)

for the inner scheme and the boundary scheme. (It is also sufficient to check
one boundary at a time.) Substituting (33) into (12) and (13) one finds that
^i k satisfies a constant coefficient second-order difference scheme whose
solution is

, =,k^l(Jp6x)^ (3^)

Actually there are two possible y's, but it is readily verified that only one
of them satisfies |p| < 1 for |g| > 1; and, therefore, it is not a valid
solution for the quarter-plane problem.

40

In Appendix A we show that there exists a solution of the form (3A) to
(12), (13), and (20) such that |g| > 1 and |y| < 1 If (29) Is not
satisfied. This proves that the scheme is unstable. By instability here we
mean that unbounded solution occurs after a fixed number of time steps for any
mesh — it precludes the possibility of reaching steady state.

It should be noted here that VN analysis of the boundary scheme does not
predict the existence of solutions of the form (33) with |g| > 1. In fact,
Gottlieb and Turkel [15] gave an example of a boundary scheme (Scheme VI, p.
184 of Reference 15) coupled with a variant of MacCormack's scheme in the
interior which is conditionally stable, yet the VN analysis of the boundary
scheme shows unconditional instability. However, Goldberg and Tadmor showed
that for a dissipative interior scheme (i.e., amplification — factor modulus
bounded away from unity for all nonzero modes) VN stability of the boundary
scheme excludes the possibility of an eigenvalue or a generalized eigenvalue.
By an eigenvalue we mean a solution of the form (3A) with |g| > 1 whereas a
generalized eigenvalue is G such that |g| = 1. Thus, if the condition
stated in (29) is satisfied no eigenvalue or generalized eigenvalue exists.
In Appendix A we show it directly. The theory of Gustafsson, Kreiss, and
Sundstrom [10] (see also, Osher [11]) states that for a system of first-order
hyperbolic equations stability is assured if there is no eigenvalue or
generalized eigenvalue. While their theory does not apply directly to the
equation

it can be modified to include this case.

41

As a concluding remark we should note that stability here implies
convergence in the sense of Lax — the numerical solution converges to the
analytic one as the mesh size tends to zero for fixed time t. This is
clearly only a necessary requirement to reach steady state.

Two-Dimensional Numerical Results

A limited number of numerical tests for cases involving stretched grids
and nonlinear transonic flow have convinced us that the discontinuous-a
schemes (e.g., Eq. (32)) are not as reliable as the scheme using Eq. (20) with
restriction (29). Some numerical results are presented in Tables 1 and 2. In
the tables, the following identification is used for the various boundary
schemes:

Scheme I: Original TAIR scheme; Eqs. (13) and (15) at boundary, with a
increased at 3 lines adjacent to boundary to satisfy restriction
(28) with 10% safety margin.

Scheme II: Exact annihilation of boundary residual; Eqs. (15) and (32),
with 0) = 1 at boundary only.

Scheme III: Eq. (20) and restriction (29) with 10% safety margin.

Table 1 shows a series of numerical tests for incompressible flow over a
circle, with varying degrees of mesh stretching near the boundary. The TAIR
code was used with oi = 1.8 at all points except as noted in schemes II and
III, and with the default settings for the a-cycle (a ^^^ = 0.07, a ^^^^ =
1.5). The mesh contained 101 points uniformly spaced around the circle and 21
points in the radial direction with stretched spacing. The first column lists
the cell aspect ratio at the boundary, Ax/Ay (= b. ) , for each case., The next

42

three columns show the number of iterations required to decrease the starting
residual by 10"'* for three schemes previously discussed. Divergence is
indicated by an entry "D." It is seen that scheme III is significantly less
sensitive to grid stretching in the normal direction than are the
discontinuous-a schemes, I and II.

Table 1. Number of Iterations to Reduce Residual by 10
Incompressible Circle Flow, 101 by 21 Mesh

Ax Scheme

Ay

II III

0.5 44 43 34

1 72 36 51

10 68 43 47

20 99 53 48

100 212 D 34

1000 400 D 127

As previously mentioned, the empirically-developed default settings in
the TAIR code provide for an Increased value of a near the surface; the
default value satisfies the restriction (28) only for the first case in Table
1, Ax/Ay = 0.5. For that case, convergence is obtained; the scheme diverges
for the other listed cases for which the default setting violates restriction
(28). In scheme I, the value of a near the surface met the restriction, and
convergence was obtained for all the listed cases.

43

It should be noted again that the stability analysis presented herein is

valid only for subsonic flow, when the AF2Y scheme is guaranteed to be

hyperbolic in time. When the flow becomes locally supersonic, the linearized

Eq. (1) will have a < 0, and a term which simulates i, must be added for

^xt

stability [16]. The effect of including such a term (e.g., in the second
factor of Eq. (9)) has not been studied at present. With that cautionary
remark, we present results for transonic cases in the next table.

Table 2 presents results for two transonic cases for an NACA 0012
airfoil: (1) Zero incidence with free-stream Mach number M = 0.85 and

00

(2) 2° incidence with M^ = 0.75. All cases were run with cj = 1.8, but with

different a- cycles. It can be seen that there is little difference in the

convergence rate among the schemes, except that scheme II is noticeably slower
than schemes I or III for case (2).

Table 2. Number of Iterations to Decrease Residual by 10 for

Transonic Flow. NACA 0012, Default TAIR Mesh, 149 by 30

Flow Condition ^""^^"^^

II III

M = 0.85

00

Zero incidence

190 174 187

M = 0.75

00

2° incidence

190 360 226

44

Three-Dimenslonal Version of AF2Y

A three-dimensional (3-D) version of the AF2Y scheme is presented in Ref.
7. It is different from the 2-D version discussed up to now, in that the
factors are reversed in order. That is, the scheme can be written in the
present context as:

^« - fj ^zK^2 - f ^xx)^« - h^y)^*jU -^^"u -^ «^(« - ^^)^*j,k-l,£ (35)
where

Because the factors are reversed, we will refer to this scheme as AF2YR.

Here the third coordinate direction is z, which can be thought of as the
spanwise coordinate for a wing. The x- and y-coordinates are still the
streamwise and normal coordinates as in the 2-D problem. The boundary
operator corresponding to Eq. (8) is:

S*j,l,£ = ^-^x ^ 2bJy + c6^J *j^^^^ - 2bAys(x). (37)

The scheme is implemented in three stages, as follows:

1. (a - ^ 6 ) g.„ = acjLi})" „ + ab_f . , , „ (38)

^ b_ zz-* ^j£ ^jk£ 2 j,k-l,£

2- (^2 - f ^xx) fjk£ = ^n (35)

3. (a - b^Jy) A^.^^ = fjj^^. (40)

45

The solution for f proceeds in planes, outward from the wing surface, using
the tridiagonal Eqs. (38) and (39). The third stage (40) proceeds inward,
solving for the correction in a bidiagonal march.

Again, the main problem is how to initiate the first stage. In Reference
7, the boundary condition used for f is

^j.o,. = «• (^1)

We can again consider the more general boundary conditions studied previously,

'j.o.n-^^j.i.t («>

or

^j,0,il =-BJ Vj,l,il (^3)

corresponding to Eqs. (18) and (30), respectively. Actually, condition (42)
can only be approximately modeled in the 3-D problem, with some splitting
error in the first two factors. That is, we can approximate Eq. (42) by
solving, at k = 1:

Equation (43) is easily implemented by replacing o) in Eq. (38) by o) + 3
and setting f. „ = 0.

46

stability Analysis of the 3-D AF2YR Scheme

A VN analysis of the 3-D interior scheme shows that VN stability is
achieved under restrictions (16) and (17). VN analysis of the boundary scheme

(42) shows that sufficient conditions for stability of the VN boundary scheme
are:

< u) < 1 - Y (46)

and

Y < 1. (47)

The same criteria are obtained in the 2-D counterpart of the AF2YR scheme with
boundary condition (18). The corresponding criteria for boundary condition

(43) are:

< 0) + e < 1. (48)

At this time we have no numerical experiments to test the stability and
convergence of the 3-D boundary conditions (42) or (43) and the criteria (46)
or (48). However, some comments about the use of AF2YR versus AF2Y are in
order.

In the AF2YR scheme, the use of boundary condition (42) or (43) makes the
scheme parabolic at the surface; i.e., the time-like equation at the boundary
is:

0^^ = V^({) (49)

where

a = b2 (1-y)/co (50)

47

for Eq. (42), and where

= b2/(a3 + 3) (51)

for Eq. (43). In the case of AF2Y, the boundary equation remains hyperbolic ,
like the interior scheme, with

- a<t>yt = "7^* (52)

where

a = h^lQ., (53)

It Is felt that for this reason AF2Y may lead to faster convergence. It would

appear that there is no difficulty in implementing such a scheme in 3-D, as:

1. (a + b^Jp f .^^ = ao^L^.j^^ (54)

2. (ab2-c6 Jgj, = V + «V<^j,k+l., (55)

3- (l - -4- 6 1 A<j)., „ = g.„. (56)

■^ ab- xx^ jk£ ^j£. ^ '

The factors in the second and third stages could also be interchanged. The
first stage is initiated by using Eq. (20), and the same stability and
restrictions (26) and (27) hold.

48

CONCLUDING REMARKS

We have studied the stability of the AF2Y scheme with several boundary
conditions for the intermediate variable. The von Neumann method provides a
useful tool for this study in view of the Goldberg-Tadmor theorem, and the
results were verified in the two-dimensional case by the more complete GKSO
theory.

In general, the boundary schemes place a limitation on a and oi which
is more restrictive than the requirements for the interior scheme. Since
small ct is desirable to damp low-frequency errors, one strategy involves
increasing a at or near the boundary to meet the boundary restriction while
using smaller a in the interior mesh. Such "discontinuous-a" schemes
require further analysis of the stability at the line next to the discon-
tinuity since the scheme there is no longer the interior scheme. They diverge
on certain stretched grids. A safer strategy is to decrease o) at the
boundary to conform to the restrictions. This results In a more robust
scheme; and it does not appear to suffer much, if any, loss in convergence
rate.

In regard to the 3-D AF2Y scheme, the current implementation in the TWING
code involves a reversal of the factors from the 2-D TAIR code. We refer to
this scheme as AF2YR. Although the reversal of the factors makes
no difference in the interior (for the linear constant-coefficient case),
there is a significant difference at the boundary. The AF2YR boundary scheme
is parabolic in time as opposed to hyperbolic for AF2Y. For this reason,
there may be a preference for the AF2Y, as in the TAIR code.

49

APPENDIX A

Application of the GKSO Theory to the AF2Y Scheme

In the GKSO theory [10], [11], the interior and boundary schemes are
considered as a coupled problem. Instead of substituting the Fourier
solutions as in Eq. (22), the class of trial solutions is extended to

,n -,n i( jpAx) k . . , .
({).j^ = G e ^-^^^ 'u (Al)

where p is a complex number not restricted to lie on the unit circle in the
complex plane. Fourier modes are retained in the direction tangential to the
boundary under study. The trial solutions are substituted into the interior
and boundary schemes, Eqs. (9) and (20), to obtain, respectively:

[a + b^(l - -)] [- ah^(]i - 1) + 2A](G - 1) = ao3[-2A + b(u - 2 + -)] (A2)

and

[- ab (y - 1) + 2A](G - 1) = n[-2A = 2b(y - 1)], (A3)

where Eq. (8) for L (})? is used for the right-hand side of Eq. (A3)

and

where we have used the notation of Eq. (24).

Equations (A2) and (A3) are two simultaneous equations for the unknowns
G and y. In the theory, we are concerned only with values of \i inside
the unit circle, i.e., only those solutions which decay away from the

50

boundary. If the solution of Eqs. (A2) and (A3) yield G > 1 for p < 1,
the scheme is unstable.

If Eq. (A3) is divided into Eq. (A2), G is eliminated; and there results
an equation for y :

n[ay + bj(y - 1)][-2A + 2b (y - 1)] = au)[-2yA + b(y - 1)^]. (A4)

First, it will be shovm that for

A = a(l - cos 5) = ,

there is a value of y inside the unit circle. When A = 0, Eq. (A4) reduces
to two linear factors:

(y - l){[2f^ (a + bj) - aojjy - 2h^ U + ato} = 0. (A5)

The root y = 1 is a solution of Eqs. (A2) and (A3) only when n = o),
corresponding to y = 1. (See Eq. (21).) Then

G = 1 - 0) (A6)

and the restriction (16) must be satisfied. The other root is:

2b, J2 - au)
IQ. (a + b.) - ato

which is less than 1.0 and is arbitrarily close to 1.0 as a approaches zero.

51

Using Eq. (A3), we can show that for any complex y such that its real

part is less than 1, G < 1 if and only if restrictions (26) and (27) are
satisfied. Thus, let

y = Pr + iUj (A8)

where p and li^ are the real and imaginary parts. Substitution of Eq,

(A8) into (A3) and multiplication by b|^ gives:

[ab (1 - y^) + 2Ab^ - iabuJ(G - 1) = - 2nbjA + b (1 - y^) - iby^.]. (A9)

2
The condition G < 1 then yields:

n{A^bj(2 - n) + Ab(l - yj^)[a - Qh^ + (2 - n)bj

+ b2(a - nb^)[(l - y^)^ + y^]} > 0. (AlO)

For Pn < 1, the restrictions (26) and (27) are sufficient to ensure the
inequality (AlO) for arbitrary positive values of A, bj^ , and b, regardless of
the magnitude of y_. When A = 0, a value y < 1 always occurs, as shown
by Eq. (A7); and the scheme will be unstable unless restriction (27) is
satisfied. Thus, restriction (27) is necessary; and when it is satisfied (for
small a), restriction (26) will also be satisfied.

Acknowledgment

We thank Dr. Terry Hoist of NASA Ames Research Center for supplying the
original TAIR code and suggesting the circle test case and Dr. Eitan Tadmor of
ICASE for discussions of the Goldberg-Tadmor theorem.

52

REFERENCES

[1] D. W. Peaceman and H. H. Rachford, Jr., "The Numerical Solution of
.Parabolic and Elliptic Differential Equations," J. Assoc. Comput. Mach .,
Vol. 8, 1955, pp. 359-365.

[2] W. R. Briley, "Solution of the Three-Dimensional Compressible Navier-
Stokes Equations by an Implicit Technique," Proceedings of the 4th
International Conference on Numerical Methods in Fluid Dynamics , 1974.

[3] R. M. Beam and R. F. Warming, "An Implicit Factored Scheme for the
Compressible Navier-Stokes Equations." AIAA J ., Vol. 16, April 1978,
pp. 393-402.

[4] W. F. Ballhaus and J. L. Steger, "Implicit Approximate-Factorization
Schemes for the Low-Frequency Transonic Equation," NASA TMX-73082,
November 1975.

[5] W. F. Ballhaus, A. Jameson, and J. Albert, "Implicit Approximate-
Factorization Schemes for the Efficient Solution of Steady Transonic
Flow Problems," AIAA J ., Vol. 16, June 1978, pp. 573-579.

[6] T. L. Hoist, "An Implicit Algorithm for the Conservative, Transonic Full
Potential Equation Using an Arbitrary Mesh," AIAA J . , Vol. 17, October
1979, pp. 1038-1045.

53

[7] T. L. Hoist, and S. D. Thomas, "Numerical Solution of Transonic Wing
Flow Fields," AIAA Paper 82-0105, January 1982.

[8] A. Jameson, "Acceleration of Transonic Potential Flow Calculations on
Arbitrary Meshes by the Multiple Grid Method," Proceedings of the AIAA
4th Computational Fluid Dynamics Conference , Williamsburg, Va., 1979,
pp. 122-146.

[9] D. L. Dwoyer and F. C. Thames, "Accuracy and Stability of Time-Split
Difference Schemes," Proceedings of the AIAA 5th Computational Fluid
Dynamics Conference , Palo Alto, California, pp. 101-112.

[10] B. Gustafsson, H.-O. Kreiss, and A. Sundstrom, "Stability Theory of
Difference Approximations for Mixed Initial Boundary Value Problems II,"
Math. Comput ., Vol. 26, 1972, pp. 649-686.

[11] S. Osher, "Systems of Difference Equations with General Homogeneous
Boundary Conditions," Trans. Amer. Math. Soc , Vol. 137, 1969, pp. 177-
201.

[12] F. C.Dougherty, T. L. Hoist, K. L. Gundy, and S. D. Thomas, "TAIR - A
Transonic Airfoil Analysis Computer Code," NASA TMX-81296, May 1981.

[13] J. A. Trapp and J. D. Ramshaw, "A Simple Heuristic Method for Analyzing
the Effect of Boundary Conditions on Numerical Stability," J. Comput.
Phys . , Vol. 20, 1976, pp. 238-242.

54

[14] M. Goldberg and E. Tadmor, "Scheme-Independent Stability Criteria for
Difference Approximations of Hyperbolic Initial Boundary Value
Problems," Math. Comput ., Vol. 36, April 1981, pp. 603-626.

[15] D. Gottlieb and E. Turkel, "Boundary Conditions for Multistep Finite-
Difference Methods for Time-Dependent Equations," J. Comput. Phys ., Vol.
26, 1978, pp. 181-196.

[16] A. Jameson, "Iterative Solution of Transonic Flows over Airfoils and
Wings, Including Flows at Mach 1," Comm. Pure Appl. Math ., Vol. 27,
1974, pp. 283-309.

55

MULTIPLE STEADY STATES FOR CHARACTERISTIC
INITIAL VALUE PROBLEMS

M. D. Salas
NASA Langley Research Center

S. Abarbanel

Tel-Aviv University, Tel-Aviv, Israel

and

Institute for Computer Applications in Science and Engineering

D. Gottlieb

Tel-Aviv University, Tel-Aviv, Israel

and

Brown University

Abstract

The time dependent, isentropic, quasi-one-dimensional equations of gas
dynamics and other model equations are considered under the constraint of
characteristic boundary conditions. Analysis of the time evolution shows how
different initial data may lead to different steady states and how seemingly
anomalous behavior of the solution may be resolved. Numerical experimentation
using time consistent explicit algorithms verifies the conclusions of the
analysis. The use of implicit schemes with very large time steps leads to
erroneous results.

Research was supported in part by the National Aeronautics and Space
Administration under NASA Contract Nos. NASl-17070 and NASl-18107 while the
second and third authors were in residence at ICASE, NASA Langley Research
Center, Hampton, VA 23665-5225. The third author was also supported by AFOSR
Grant 85-0303.

56

INTRODUCTION

Consider a steady, isentropic flow In a dual-throat nozzle with equal
throat areas, and assume that the flow is choked; then it is well known [1]
that the flow between the throats can be either completely subsonic or
supersonic depending on the initial state of the flow and the path taken to
reach the steady state. If we experiment numerically with the above problem
using either the isentropic quasi-one-dimensional gas dynamics equation or
some "simpler" model equation, then some of the results obtained are rather
peculiar.

(1) If the initial data correspond to sufficiently high supersonic flow (or
sufficiently low subsonic flow), then the steady state flow obtained
between the two throats is indeed completely supersonic (subsonic).

(2) If the initial data are completely supersonic (or subsonic), but below a
certain level (above a certain level), then the steady state flow
contains a shock wave connecting the supersonic branch of the solution
to the subsonic branch. For the model equations considered, the shock
corresponds to an isentropic jump, and its location depends on the
initial data.

(3) Results (1) and (2) above are observed when time accurate schemes are
used. However, the implicit backwards Euler scheme with large time
steps yields steady states that are not reachable through a time
accurate path from any class of nontrlvial initial conditions. These
steady states include not only discontinuous solutions (as observed in
[2]), but also unstable smooth solutions.

57

(4) The numerical treatment of boundary conditions is very important in
obtaining the proper results. For example, with central space
differencing one may have a stable algorithm that does not converge in
time to a steady state if the sonic conditions are invoked in order to
supply numerical boundary conditions.

The purpose of this paper is to present our findings, and to provide, where
possible, a mathematical explanation of the observed behavior, thereby
removing the apparent peculiarities. We will show that the nonuniqueness
aspect of the steady state solution is a by-product of the fact that the
boundary conditions for the evolution equations are prescribed along
characteristic curves. This is true for the dual throat problems due to the
sonic conditions imposed at the throats. The model problems were therefore
chosen to show this behavior.

In Section 2 we study the model equation

2
9u 9 ^u >> _ , .

The relevance of this model equation to the quasi-one-dimensional gas
dynamical equations is somewhat peripheral. However, it is rich in the number
of possible steady solutions that it admits, including unstable continuous and
discontinuous solutions. In this section we discuss the proper way to
formulate the characteristic boundary conditions for first order quasi-linear
hyperbolic equations.

58

In Section 3 we consider the model equation

2
8u 3 (-u >> .

This model equation has solutions which qualitatively behave like those of the
isentropic dual throat nozzle problem. The simplicity of the model, however,
affords a detailed study of the possibilities for anomalous behavior. This
model equation will also show us how to quantify such vague terms as
sufficiently high (or low) supersonic (subsonic) initial conditions that were
mentioned in (1) and (2) above. These results are summarized in Theorems 1
and 2.

In Section 4 a model scalar equation is developed which has all of the
interesting physical aspects of the complete isentropic quasi-one-dimensional
gas dynamic equations governing the dual throat nozzle problem. To develop
this equation, our guideline was to retain the differential equation
exhibiting the characteristic boundary condition and to model the other
dependent variable by assuming constant total enthalpy during the time
evolution. By comparing the theoretical results of the model equation to
numerical calculations for the complete system of equations, this section
shows that the proposed single equation is indeed a good model of the complete
system. Here, by the "goodness" of the model we mean that all of the
important features of the system are retained.

Recently Kreiss and Kreiss [4] have investigated the above model
equations in the presence of a linear dissipative term of the form eu

XX

They show that in this case the solution is unique and discuss the convergence
properties of their numerical scheme.

59

2. FIRST EXAMPLE

Here we consider the scalar hyperbolic partial differential equation

If "'lirff') = "^1 -")' < X < 1, t > 0,

(2.1)
u(x,0) = g(x).

For reasons mentioned in the introduction, and to be discussed in detail in
Section 4, we are interested in cases that model physical situations in which
the boundaries are characteristic. In practice, when (2.1) is solved
numerically as a characteristic boundary value problem, the boundary
conditions are imposed dynamically as follows:

if u(£Q,t) > (£q = Ax)
u(0,t) = { (2.2a)

unspecified if u(e^,t) ^

if u(e^,t) < (e^ = 1 - Ax)
u(l,t) = { (2.2b)

unspecified if u(e, ,t) >^

There are two families of continuous steady states satisfying (2.1) and the
analytical versions of (2.2):

u = (2.3)

u = 1 - e'^"^ (0 < n < 1). (2.4)

60

The stability theory of ordinary differential equations applied to the
characteristic equation du/dt = u(l - u) easily shows that the steady state
solution u = Is unstable.

There are also weak solutions connecting various branches (different n's)
of (2.4). These discontinuous solutions are unstable as will be demonstrated
now. Let

Uj^ = 1 - e e (2.5)

be a steady state corresponding to n = Hi >

^2 -X
u„ = 1 - e e (2.6)

K

be another branch.

Since we want to rule out "expansion shocks," i.e., discontinuities that do
not obey the "entropy condition" u^ > > Uj^, we will consider only the case
of 1 >^ Ho > ri , >^ 0, although the analysis is unchanged if ti« < ri, . For a
steady state shock we require u^(x„) + u_(x_) = 0. This determines the shock

Lb Kb

location, xg, to be

^^1 ^ ^^2
X = £n 2 • (2.7)

We now ask, what will be the shock speed, x_ = y (u^ + u ), if Xg is
perturbed to Xg + £ ? Upon substituting the perturbed shock position in
(2.5) and (2.6), we get for the new shock speed

61

-i^5— ^ = 1 - e ^ « e + OCe"^). (2.8)

Thus, if e > (e < 0) the shock will move to the right (left), showing that
the solution with a shock is not stable.

We have thus shown that in the steady state we need consider only the
smooth solutions in (2.4). We will now demonstrate that these solutions are
reachable from initial data. The demonstration is first done for the case
n = 0, g(x) > for all x > 0, and g(0) = 0.

Consider the problem (2.1), and let

g(x) = b(l - e~^), b > 0. (2.9)

The solution to this problem is readily verified to be

1 "^

u(x,t) = b ^ —. (2.10)

e "^ + b(l - e""^)

Clearly, as t ■»■ <», u(x,t) -»■ 1 - e , which is a proper steady state.

Suppose now g(x) is not a multiple of the steady state but is a general
initial condition still satisfying g(0) = 0, g(x) > 0. The characteristic
equations are

^=u (2.11)

^= u(l - u). (2.12)

From (2.12) one gets

u iii^ _ (2.13)

g(5) + (1 - g(0)e~^

62

where C = 5(x,t) is the origin of the characteristic passing through x
and t. By inserting (2.13) in (2.11) and integrating again along the
characteristic, we get the following implicit relation between C, x and t:

e^-^-^ = [g(0 + (1 - g(0)e"'] (2.14)

or, upon rearranging

g(0=^- -. (2.15)

e - 1

The argument is now as follows: x - C is finite (0 < x-5 < 1), and thus as
t •> ", g(5) -»■ 0, but giO ■»■ only for 5 •> 0. Hence, for any finite x,
as t increases, g(5) takes the large time asymptotic form of

g(0 --^^^^ (t » 1). (2.16)

e - 1

Substituting (2.16) in (2.13) we get

u(x,t) ~ ^ " ^ (t » 1). (2.17)

1 - e

Thus, as t •»• ", u(x,t) •> 1 - e '^ regardless of the detailed form of the
initial data.

For other types of initial data (e.g., g(x) = for some x = xg), the
proof is the same with n = x„ and the coordinate x transformed to

X X "" ^A *

If g(x) has several simple zeros, then the interval £. ^ £. 1 ^s sub-
divided by the zeros. Their relative locations will determine the proper n.

63

In particular, If g(x) Is a periodic function, g(xJ = 0, with
Xj = ^ . J = 0,1...,N, then

If

or

(1) sgn g'(0) = sgn g'(l) >
(11) sgn g'(0) = - sgn g'(l) <

(2.18a)

and

If

or

(1) sgn g'(0) = sgn g'(l) <
(11) sgn g'(0) = - sgn g'(l) >

(2.18b)

(where primes denote differentiation with respect to the Independent
variable). In summary, this example demonstrates the richness of possible
steady state solutions.

(1) There Is an unstable smooth solution, u = 0.

(2) There are unstable discontinuous solutions.

(3) There Is a one-parameter family of smooth steady states,

u = 1 - e^~^

with the value of the parameter depending only on the Initial data, a
direct consequence of the problem having characteristic boundary values.

64

It is interesting to note that if the right-hand side of equation (2.1)
is taken to be u(u-l), instead of u(l-u), then there is only one possible
stable steady state solution satisfying the boundary conditions (2.2), namely

u = 0.

Note that this was one of the unstable solutions of the previous case.

2.1 NDMERICAL RESULTS FOR THE FIRST EXAMPLE

2.1.1 Explicit Form

The conservative, upwind, first order scheme of Engquist-Osher (E-0),
[3] is used to approximate the hyperbolic system of conservation laws
represented by

If - |j (^) = M.,u) (2.19)

where h is a source term. Let u^ represent the discrete value of u at
t" = nAt and x, = iAx. The explicit E-0 scheme for equation (2.19) is,

n+1 n 1 At
i i 2 Ax

n x2

T(i-^i.l)("i.i)'-^V"i>'-T<^^ Vi>K-l^

+ h(iAx,uJ)At (2.20)

where the switch function 6, is defined by

65

(°

n

6, =

)

i

)>■?

'-

u'' t

\ 1 Hi

i

\ Uj

(2.21)

As usual, At satisfies the Courant-Friedrichs-Lewy condition,

At<— ^^, (2.22)

max|u I

and Ax = L/lOO, where L is the length of the interval of interest. For the
explicit E-0 scheme convergence was established according to the criterion

maxlu""*"^ - u^l < 1. X 10~^. (2.23)

i ^

The relation given by (2.23) is equivalent to requiring the steady state
operator of (2.20) to be less than 10"^. Figure 1 compares the exact and
computed steady states for equation (2.1) with initial conditions*

g(x) = - sin 2Trx. (2.24)

Note that the steady state satisfies the condition (2.18bi) and that the
initial conditions and steady state solution are such that no boundary

Note that, because of the first order accuracy of the Engquist-Osher
scheme. Figure 1 shows a slight discrepancy between the analytic and numerical
solution. The same problem run with x = 1/1000 gives results that, on the
scale of Figure 1, are indistinguishable from the analytic results. This
comment holds for all other numerical experiments, where, in order to save
computer time, we used 100 mesh points.

66

conditions are imposed at either end of the interval. The same steady state
is also obtained with

g(x) = -x(x-l)(x - 2" )•

(2.25)

Figure 2 compares the exact and computed steady states for initial conditions

g(x) = sin 2irx.

(2.26)

The steady result is in agreement with the condition (2.18ai).

2.1.2 Implicit Form

The slow convergence to steady state characteristic of explicit schemes
has stimulated research into various acceleration techniques. One of the most
promising avenues for acceleration consists of recasting the discrete equation
in implicit form. If we define the increment in time of u by

i ^ "i ~ "i'

(2.27)

then the E-0 scheme in implicit form is

i (' - h»K.l '"!« * (If * «1 "i - (f)" '■=)'"! -U'* «i-l)Vl ^1-1

1

J (1 - «i+l)(^+l)' -^ hi\f -J^'^ h-OK-lf "■ MiAx.uJ)Ax.

(2.28)

67

where 6, is defined as before by equation (2.21). To obtain equation

2
(2.28), terms of order Au. and higher are neglected. It is easy to see, by

comparing equations (2.20) and (2.28), that the right-hand side of equation

(2.28) is the steady state operator. For the implicit E-0 scheme convergence

was established by requiring that the steady state operator be less than 10"^

at all mesh points.

Figure 3 shows the steady state solution obtained using the implicit

E-0 scheme with

g(x) = sin 2ttx (2.29)

and using infinite Courant number (— = O). The steady state obtained with
the implicit form of the scheme corresponds to one of the unstable solutions
of equation (2.1). The stable solution, for g(x) corresponding to equation
(2.29), was shown in Figure 2. The peculiar behavior of the implicit
algorithm at large Courant numbers is further demonstrated in Figure 4 for

g(x) = - x(x - l)(x - j) (2.30)

and infinite Courant number. For this case, the steady state reached by
(2.28) consists of a combination of stable and unstable steady, piecewlse
solutions of equation (2.1).

68

Figure 1. Exact and computed steady states for equation (2.1) with
initial conditions (2.24) using a time accurate scheme.

69

-.5 -

u

-1.0

-1.5 -

-2.0

Figure 2. Exact and computed steady states for equation (2.1)

with initial conditions (2.26) using a time accurate scheme.

70

O Computed
— Exact, unstable

Figure 3. Exact and computed unstable steady states for equation (2.1)
with initial conditions (2.29) using an implicit scheme with
large Courant number.

71

.5r

u

O Computed

OO

oo

h^oo

-.5

oo

o

o

oo-o

o

o

o o

80

o

oO

J L

J L

J L

.1 .2 .3 .4 .5 .6 .7 .8 .9 1.0

X

Figure 4. Computed steady state for equation (2.1) with initial
conditions (2.30) using an implicit scheme with large
Courant number.

72

3. SECOND EXAMPLE

We now shift our attention to another advection problem. The steady
states of this problem are of a completely different nature than of those
found in the previous example.

The partial differential equation under consideration is

2

1^ + I- (-^l = sin X cos X, < X < TT, t > (3.1)

dt 3x ^2 -^ — —

U(X,0) = g(x), g(0) = g(TT) =

with boundary conditions as given by (2.2).

Here we have two smooth steady state solutions,

u = sin X

(3.2a)
u = - sin X.

There is also' an infinite number of possible discontinuous solutions of the
form

u = u X < X

(3.2b)
U = U X > x_

where Xg, the "shock" location, is an arbitrary point in the interval
(0,ir). Note that, in the steady state, the "shock" speed u = (u + u )/2
is zero for any < x < it and, therefore, (3.2b) is a legitimate steady
state solution. In the above solutions we have already eliminated weak
solutions that violate the "entropy condition," u^ > > Uj^.

73

We now ask two questions:

(i) From what class of initial conditions, if any, can either of the two
smooth solutions, (3.2a), be reached and
(ii) Under what circumstances is a steady shock established, and can its
location be predicted?

Consider first the two questions in the particularly simple case when

g(x) = e sin X, (3,3)

i.e., the initial data are proportional to a smooth steady state. For
3 > 1, Theorem 1 shows that the steady state is the smooth solution u =
u"*". For

3 < -1, a corollary of Theorem 1 leads to u = u~.

Theorem 1 : The solution of equation (3.1) with boundary conditions
(2.2), initial conditions (3.3) and g > 1 satisfies

lira u(x,t) = sin x.

Proof: The characteristic equations resulting from (3.1) are

dx

dt = "

(3.4)

du ^ du ^ dF 1 _,_2

dt dx dx ' 2

= u-^ = -r^. F=-y sin" X. (3.5)

74

Again using E, = 5(x,t) to designate the origin of a characteristic curve
passing through (x,t), we integrate (3.5)

ju^ -jg^io = F(x) - no

or

u = ±[2F(x) - 2F(5) + g^(5)]^^^ . (3.6)

As t ->- 0, 5 -»■ X and we have to choose the positive branch of (3.6) because
6 > 1. Thus, using F = (1/2) sin^ x,

u = [sin^ x + (e^ - Dsin^ 5]^^^. (3.7)

We claim now that for t large enough there is a unique correspondence
between a point (x,t) and 5(x,t). In fact, if a shock wave were to appear
at a certain time t > 0, it will, because of (3.7), separate two positive
states. The shock wave will have a positive speed and consequently will
propagate out of the domain. Therefore, for t large enough, we may
substitute (3.7) into (3.4),

' - /" — '' ,. ,.,» »•«

5 [2F(y) - 2F(5) + g^(5)]"^

or

X

t = / — 2 r^ 2 — m ^3-^>

C [sin^ y + (e - Dsln^ C]'^^

For every x < it , the integrand in (3.9) cannot become singular except at the
lower limit y = ? , ? -v 0. Thus, t -»■ «> as E, -*■ and the only possible
solution for very large time is, from (3.7),

75

u ^ [2F(x) - 2F(5) + g-{0]^'^ = [2F(x) - 2F(0) + g^O)]^^^ = sin x,

5^0

which completes the proof.

Corollary: Suppose that 3 in (3.3) satisfies 3 < -1 , then

lim u(x,t) = - sin x.

t-Voo

Note that in view of (3.8) the results of Theorem 1 hold for any initial
conditions g(x) such that g(0) = 0, g(x) > sin x. The corollary is thus
also valid for any g(x) < - sin x.

Still continuing with the case of g(x) = 6 sin x, we now consider

< e < 1. (3.10)

Here the steady state will be of the form (3.2b). We will show, however, in
Theorem 2 that the shock location depends on the initial condition.

Theorem 2 : The solution of equation (3.1) with boundary conditions
(2.2), initial conditions (3. 3), and < 6 < 1 satisfies

u = sin X, < X < x„

lim u(x,t) = { (3.11)

t->-<»

u = -sin X, Xj, < X < IT

76

where

Xg = TT - sin"^ / 1 - e^ > J . (3.12)

Proof; From the characteristic equation (3.5), with < g < 1, we get

u(x,t) = ±[sin^ X - (1 - e^)sin^(c(x,t))]^''^. (3.13)

In the interval (it - x , x ), x as defined in (3.12), u(x,t) cannot change
sign because the radical in (3.13) cannot vanish in said interval. Since as
t ->■ 0, u(x,t) is positive, we conclude that

u(x,t) = [sin^ X - (1 - e^)sin^(5(x,t))]^''^, tt-x < x < x . (3.1A)

In this interval the first characteristic equation (3.4) becomes

t=/'' % 2 172 ^^'^^^

? [sin^ y - (1 - 6^)sin^(5(x,t))]'^^

since t > we must have 5 < x when Tr-x„ < x < x„. As t ^ ~, C(x,t)
must therefore vanish in the limit. It is thus established that

lim u(x,t) = sin x, (tt-x < x < x ). (3.16)

t-»-oo

Next consider the interval [0,Tr-x ). Formally as t ->■ «>, in this leftmost
interval, C(x,t) must converge either to zero or ir. However, any
characteristic passing through (x,t) in the interval [0,TT-Xg) cannot
emanate from any C > x_ because this would mean a negative slope, and hence

77

a negative u in the interval (tt-x , x ); this contradicts (3.16). Having

established that 11m 5(x,t) = 0, we notice that formally It is possible for

t->-«

a characteristic curve, originating in the Interval [0,tt-x ), to start with a
positive slope (required as t ->■ 0) and change slope in the Interval. This,
however, will result in a solution containing a "shock" that violates the
"entropy condition" ul > > uj^. We thus have our next intermediate result

11m u(x,t) = sin x, (0 £ x < x ). (3.17)

t-f=o ^

It now remains for us to show that in the Interval x < x < tt the solution
must be negative and hence equal to - sin x.
We first Integrate (3.1) to get

g ir TT 2

T— / udx = - / (^) xdx.

^^0 , ^

Suppose that at the point 0<Xj<X2»"<x <Tr, u(x,t) is discontinuous,

2 + 2 —
since we admit only "shock" discontinuity u (x ) > u (x. ). Thus,

1^ / u(x,t)dx=i u^O,t) - I (u^ (x^) - u^x^)) - u^(Tr,t) (3.18)

from (3.13), u^(0,t) = and therefore,

/ u(x,t)dx < / u(x,0)dx = 2$. (3.19)

Let X be the point in which u(x,") changes sign. From (3.16), we have

78

and from (3.19) we have

X < X

s a

X

a t;

/ sin X - / sin X < 2B, (3.20)

X

a

thus.

-2 cos < 23

a

or

X < x^ (3.21)

as

and therefore

X = X . (3.22)

as

This completes the proof.

It should be noted that, in general, Xg gives a lower bound on the location
of the discontinuity whereas the area rule (3.19) yields an upper bound on it.

Corollary: Under the conditions of Theorem 2 with

-1 < 6 <

the solution still retains the form of (3.11) except that now

Xg = sin"^ / 1 - e^ < -J .

79

For arbitrary initial data the general behavior is that described in
Theorems 1 and 2 and their corollaries, i.e., one can get either solution
(3.2a) or (3.2b). If a "shock" is present in the steady state, the upper and
lower bounds for its location are given, for g(x) > 0, as follows:

-1 1 2 2 -1 / 1 ''^ 7

77 - sin /sin z - g (z) < Xg < 77 - sin J \ - j {\ g(7i)dn) , (3.23)

where z maximizes the expression sin^ x - g^(x). For negative initial data
the bounds are

5in~^ / sin^ z - g^(z) < Xg < sin"^ J ^ -\ if g(n)dn)^ . (3.24)

The upper bound reflects the "area rule" (see (3.18)). The lower bound is the
first point where u(x,t) can change sign. For g(x) > 0, the upper bound
becomes sharp (i.e., equals Xg), if u(77,t) = for all t.

3.1 NUMERICAL RESULTS FOR THE SECOND EXAMPLE

3.1.1 Explicit Form

Equation (3.1) is discretized using the explicit E-0 scheme given by
equation (2.20). Numerical calculations were performed for initial conditions
given by

g(x) = B sin X, (3.25)

80

where g is a free parameter such that ^ g < 1. The steady state shock

position as a function of 3 is plotted in Figure 5. The numerical results

are in excellent agreement with the theoretical prediction given by equation

(3.12). For any g > 1, the steady state obtained was u"*" given by equation

(3.2a).

If one uses an algorithm employing central space differencing (e.g.,

MacCormack's scheme), it is then necessary to supply a numerical boundary

condition. If the steady state value is used for the boundary condition, then

the numerical algorithm, though stable, falls to converge to steady state.

The reason is clearly due to the fact that the numerical boundary condition

does not allow for a flux through that boundary. As a consequence we have

(see (3.19))

■n
j u(x,t)dx = 2g

for all t, while the true steady state, u"*", requires

lim / u(x,t) = 2.
t^"

3.1.2 Implicit Form

Equation (3.1) is discretized using the implicit E-0 scheme given by
equation (2.28). Once again, numerical calculations were performed for
initial conditions given by equation (3.25). Now an additional free parameter
is

^^00 Ax

81

which is a measure of how big At is taken in the numerical calculations.
The results of these series of calculations are given in Figure 6. As
indicated in the figure, if "small" At's are taken (e 2V2 ), then the
steady state shock location calculated agrees with the theoretical prediction
of equation (3.12). However, as At increases, the steady state shock
position is found to the right of its theoretical location. For sufficiently
high values of At (small e's), the smooth solution is obtained.

82

1.0 r

P

.9
.8

.7
.6
.5
.4
.3
.2

.1

O Computed
— Theory

.5

Figure 5.

.7

x^/tt

.8

.9

1.0

Computed and predicted steady state shock position for
equation (3.1) with initial conditions (3.25) using a
time accurate scheme.

83

p

1.0
.9
.8
.7
.6
.5
.4
.3
.2
.1

£>0.5

Figure 6. Computed and predicted steady state shock position for
equation (3.1) with initial conditions (3.25) using an
implicit scheme.

84

4. A MODEL FOR QUASI-ONE-DIMENSIONAL FLUID DYNAMICS

A characteristic boundary value problem, where boundary conditions are
of the form (2.2), occurs in a double-throat Laval nozzle

X = X = 1

Figure 7. Sketch of double-throat nozzle

as shown in Figure 7. It is well known [1] that there are two possible smooth

steady solutions, with sonic conditions at the throats. Between the throats,

< X < 1, the flow can be either completely subsonic or supersonic, the exact

Mach number distribution, in each case, being dependent on the nozzle area,

A(x), where 1 < A(x) < A in (0,1), A(0) = A(l) =1.

max X » J V /

If one considers the isentropic case only, then the flow may be
described by the quasi-one-dimensional partial differential equations for the
Riemann variables.

iji = u + r- c.

(j) = u -

Y- 1 -

1/2
where u is the velocity, c = (yp/p) is the speed of sound, and y is

85

the ratio of specific heats for ideal gases. The equations are

1^ + (u + c) |i + ucF'(x) = 0, (4.1)

d L dX

|i+ (u - c) |1- ucF'(x) = 0, (4.2)

where F'(x) = dF(x)/dx = d(£nA(x)) /dx. This is a hyperbolic system whose
time evolution is difficult to describe analytically. We therefore seek a
model for this system so that with a single equation the most salient
features are retained. We will present numerical evidence that analytical
predictions resulting from this model equation agree very well with results
found by numerical integration of the original system (4.1), (4.2).

The model is derived using a single assumption, namely that the total
enthalpy is constant not only at steady state but also during the transient
phase. The mathematical expression of this assumption is that

4,2 + ^2 ^ 2(1 - a) ^^ = ,___ = _^^^ ^^ (4,3)

«4

16

2
^0

2a -

1

2

Y -

1

where

a=X^, (4.4)

Cq is the stagnation sound speed, and c* is the sonic sound speed, i.e.,
c^ is the sound speed at a sonic throat.

We now face the choice of solving (4.3) for either \|> in terms of ^,
or vice versa. This dilemma is resolved by recognizing that our "physical"
problem will impose characteristic boundary conditions on (4.2), and we would

86

like our model equation to retain this feature. Therefore, (4.2) is the
relevant equation. Solving for ip gives

1 - a ^ J_
a a

2 2

4a c

2a - 1

* 2

- (2a - l)<i,

1/2

(4.5)

where the positive branch was chosen in order to satisfy the steady state
boundary condition at x = 0, i.e., at the first throat, where

^* =

2a

* 2a - 1 *

c*; <!'* = -

2(1 - a)

, 2a - 1 ^*'

(4.6)

Using (4.5) in (4.2), and defining

<t> = '^/'Pi

(4.7)

the equation (4.2) takes the form

|f- + A(;)|f = H(;)F'(x)

(4.8)

where

A(((>) = (J) +

i-:i^ /7TT2

/ 2a - 1

H(i) -K^)

1 - 2^"

-iOLjLal :/7TT

/ 2a - 1

(4.9)

(4.10)

T = tc^.

(4.11)

Notice that the time scale, t, is determined by the sonic conditions.

87

For the sake of clarity let us first examine the simple case of a = 1
(y = 3), which corresponds to the flow of products caused by detonating solid
explosives. Equation (4.8) then becomes

If- + J H = ^ (1 - 2j2)F'(x), F(x) = £nA(x). (4.12)

A smooth steady state solution of (4.12) with (j)(0) =0 is

J2(x) = j(l - e~^^^^), (4.13)

since A(0) = 1, and so, as In (3.2a) we have two possible steady states. One
is positive (supersonic) and the other is negative (subsonic):

2+ _ r A(x) - 1 ^1/2 ,, ...

* ~ ^ 2A(x) ) ('^•1'^)

:- - _ r A(x) - i a/2

* " ^ 2A(x) J • (^^-l^)

Bearing in mind the results of the previous sections, we will show that in the
time evolution problem, ^ and (^ are reachable from different initial
conditions. Clearly (4.14) and (4.15) can be connected by a steady shock -
and again, because of the symmetry of ^ and (j) , the steady shock location
Xg could be anywhere in the interval (0,1). We will show that here too
bounds on Xg can be found and compare them with results of numerical
integration of the original system (4.1), (4.2).

We will concentrate on the positive branch (4.14), showing that if the
initial condition is given by

88

:+ _ -rA(x) - 1t1/2

<t>(x,0) = g(x) = &f = g r 2A(x) ] (^-16)

with

1 < 3^ <_M2L___ ^ (4.17)

max

where \^ax ^^ ^^^ maximum area in the nozzle, then lim (j)(x,t) = (j) (x). A
solution of the second characteristic equation,

^=i^ = T(l-2;')F'(x) (4.18)

is given by

|l - 2j^l = |I - 2g2(g(x,T))|A(5(x,T))/A(x), (4.19)

where as before 5(x,t) is the origin of a characteristic curve passing
through (x,t). Since we have chosen (see (4.17)) g (x) to be smaller
than 1/2, it follows from (4.19) that

<t>(x,5) = ±

A(x) ■• A(g)ri - 2g^(g(x,T))1

fey^

1/2

(4.20)

where 5(x,t) is to be determined from the first characterisitc equation

T=/''.-^^^. (4.21)

From (4.16) we see that a positive (negative) g will initially select a
positive (negative) branch of (4.20). By an argument similar to that used in
Theorem 1, it remains for us to show that cj) thus initiated will not change

89

sign while evolving to steady state. This follows immediately from (4.20) if
we use for g(x) equation (4.16) with g > 1.

Next we consider the discontinuous steady state solution. The initial
data are now taken so that |g(x)| < {"*", see equation (4.14). A lower bound
for xg is found by inquiring about the zeros of (4.20) - the argument is the
same as in the previous section. The radical in (4.2) is zero

A(xg) = A(z)(l - 2g2(z)) (4.22)

where, as before, z maximizes the expression A(x)(l - 2g2(x)). To find the

upper bound we have to devise an "area rule" for equation (4.12). Because of

the structure of the right-hand side of (4.12), it is no longer / <},(x, T)dx

which is conserved. To find the appropriate "area rule," we divide both sides

of (4.12) by 1 - 2(j) > 0. The resulting equation after integration by x
over the interval may be written as

It /'^- ^-^^^ dx - 4- /'|-[iln(l - 2;2)]dx = J- F(x)
° 1-/2; >^2 0^- /2

^ = 0. (4.23)

A A

Under the usual area rule assumptions, (|)(0,t) = ())(1,t) = 0, we have

^ l+_/7J

J £n J. dx = const. (4.24)

1 - /? 4,

Therefore, an upper bound for xg is found from

""s , . ./T :+ 1 , . ,^ ?- 1

/ £nl-L:4Vdx./ Zn '^'lt dx = / ,n i-^^^li^l dx. (4.25)
1-/2 <|, X 1-/2 (|, 1-/2 g(x)

90

When g(x) <_ &<^ , (g < 1) we expect, as in the previous example, the upper
and lower bounds on Xg to coincide. This was indeed verified in numerical
experiments with a particular area distribution A(x).

Recalling that (4.12) is a scalar model equation representing the
systems (4.1), (4.2), we find it interesting to note that this 2x2 system also
possesses an area rule, namely:

1^/ (</'+<(. )dx =i [(,1/2(1, t) + <f2(l,t)) - (/(0,t) + <|,2(0,t))]. (4.26)
Under the assumption that <))(0,t) = ^(l,t) = 0; i|j(0,t) = ij;(l,t), we have

Iy J ('I' + <l))dx = 0. (4.27)

We can now use this to test the "goodness" of our model by comparing the shock
location predicted from (4.25) with that of the system, whose solution is
found numerically. This comparison is carried out in the next section.

Having concluded the analysis of the a = 1 case, let us now return to
the more general formulation (4.8). In particular, let us consider the case
of Y = i«4 (a = .6), corresponding to air. We next show how (4.8) may be
cast in a form similar to the "decoupled" one in (4.12). Multiply both sides
of (4.8) by r'(,j)) (r' = dr/d<j)) to obtain

A

If + r 1^ = HlMl)l F'(x) = K(r)(r^ - r)(r - r_)F'(x), (4.28)
<{i'(r)

where

91

^+ = /I . r_ = - /II . (4.30)

The quantities r_ and r+ are the values of r which, in the steady state,
correspond to Mach numbers of zero and infinity, respectively. For general
values of y, K(r), r+ , and r_ are replaced by K(r,a), r+(a), and r_(a).
K(r,a) will have the same structure as in (4.29).

It is easy to verify that K(r), given by (4.29), is a positive, slowly
monotonically decreasing function in the relevant range r < r < r . In fact
K(r_) « 2K(r^) = .309. In the case of y = 3, i.e., equation (4.12), r = <{,
and we have r^ = -r_ = 1/v/T and K(r) = constant. It is thus clear that
the topological behavior of (4.28) is the same as that of (4.12), and the
arguments carry over. In particular the non-unique smooth steady states
depend on the initial data in the same fashion with respect to 6.

4.1 NUMERICAL RESULTS FOR QUASI ONE-DIMENSIONAL EQUATIONS

Here we study numerically equations (4.1) and (4.2) for y = 2, namely:

lT^k(f)=-i('^'-*'K(-) (^-31)

2

li-^lir(f-) =t(^'-*')F'(x). (4.32)

92

The area of the dual-throat nozzle is defined by

A(x) = (^ - '^' ^ {' - '^'^ - 'ff . 0<x<l. (4.33)
2(1 - d)(l - d(2x - 1)^)^ ~ ~

where d is a parameter related to the maximum area by

A - (1 - d)^ + 1 ,, _,.

^max - 2(1 - d) • ^'''^''^

For the numerical experiments, we have used d = 1/6 which results In
\iax ~ ^•^* ^^® steady state Mach number distribution Is

M(x) = A(x) ± /a^(x) - 1 , (4.35)

and the steady state solution to (4.31) and (4.32) as a function of the Mach
number Is

I/) = /3 (1 + M)/(l + M^) ^^2 (4.36)

(t) = - /3 (1 - M)/(l + M^) ^2 . (4.37)

With the stagnation pressure and density used as reference values, the value
of -^^ Is /6".

93

4.1.1 Explicit Form

Equations (4.31) and (4.32) are discretlzed using the explicit E-0
scheme given by equation (2.20). Numerical calculations were performed with
initial conditions corresponding to

^(x,0) =3 /6 [Mg^]V2^ (,^33)

which is equivalent to (4.16), and with

'J^(x.0)=/6[A(|1^]V2, (,.39)

or

♦ (..0) = /6 (l - ^'{^^^^f^ (4.40,

The Initial conditions given by (4.39) correspond to the exact, steady
solution for t|) while those given by (4.40) correspond to conditions for ij;
consistent with (4.38) and constant total enthalpy, (4.5). The steady state
reached was the same in either case; therefore, the results reported here are
for calculations with (4.40) only.

Figure 8 summarizes the numerical results. The figure compares the
predicted steady state shock position as given by (4.25) for the model
equation (4.12) and the computed position for the system (4.31) and (4.32).
As is evident from the figure, the agreement is very good.

94

4.1.2 Implicit Form

Equations (4.31) and (4.32) are dlscretized using the Implicit E-0
scheme given by equation (2.28). Equations (4.38) and (4.40) are again used
as Initial conditions. The numerical results are summarized In Figure 9. As
shown In the figure, the steady state shock position depends on the Courant
number as measured by the parameter

E = 100 7^ . (4.41)

At

For values of e >^ 10 the steady state shock position Is the same as that
predicted by the explicit form. For values of e < 10 (large At), the steady
state shock position bifurcates at certain values of 0.

95

p

1.0
.9
.8
.7
.6
.5
.4
.3
.2
.1

Figure 8. Predicted steady state shock position given by (4.25) for

equation (4.12) and computed position for system (4.31) and

(4.32) with initial conditions (4.38) and (4.40) using a time
accurate scheme.

96

1.0
.9
.8
.7
„ .6
^5
.4
.3
.2
.1

£ = O.OK

£ = 0.03-^

£ = 0.05-^

.1 .2 .3 .4 .5 .6 .7 .8 .9 1.0

Figure 9. Predicted steady state shock position given by (4.25) for

equation (4.12) and computed position for system (4.31) and

(4.32) with initial conditions (4.38) and (4.40) using an
implicit scheme.

97

CONCLUSIONS

In this paper we analyzed several model equations for characteristic
initial boundary value problems and examined numerically these as well as the
quasi-one-dimensional isentropic Euler equations of gas dynamics.

We showed that because of the characteristic nature of the boundary
conditions the resulting steady states, whether smooth or discontinuous,
depend on the initial data. Different initial conditions may yield different
steady states. We also gave an example (see Section 2) of solution to the
steady state equation which cannot evolve from the initial data. Thus from
the point of view of the time-dependent equation, we find there are no non-
unique steady states.

Another conclusion that one may draw is that in order to have complete
confidence in the results, numerical schemes for characteristic initial
boundary value problems should be time consistent and employ only suitable
boundary conditions. Thus we have shown that implicit methods, even for
finite Courant numbers, may yield solutions which are piecewise combinations
of non-unique solutions of the steady state equations. In fact, such
numerically implicit algorithms may converge to solutions which also include
parts of unstable steady states.

98

REFERENCES

[1] L. Crocco, "One-Dimensional Treatment of Steady Gas Dynamics" in
Fundamentals of Gas Dynamics, Vol. Ill of High Speed Aerodynamics and
Jet Propulsion , Howard W. Emmons, ed., New Jersey, (1958), pp. 183-186.

[2] P. Embid, J. Goodman, and A. Majda, "Multiple Steady States for 1-D
Transonic Flow," SIAM J. Sci. Stat. Comp ., Vol. 5, No. 1 (1984), pp.
21-41.

[3] B. Engqulst, and S. Osher, "Stable and Entropy Satisfying Approximations
for Transonic Flow Calculations," Math. Comp ., Vol. 34, (1980), pp. 45-
75.

[4] G. Kreiss and H. 0. Kreiss, "Convergence to Steady State of Solutions of
Burgers' Equations," NASA Contractor Report No. 178017, ICASE Report No.
85-50, December 1985.

99

A MINIMUM ENTROPY PRINCIPLE IN THE GAS DYNAMICS EQUATIONS

is

Eitan Tadmor

School of Mathematical Sciences, Tel-Avlv University

and
Institute for Computer Applications in Science and Engineering

ABSTRACT
Let u(x,t) be a weak solution of the Euler equations, governing the
inviscid polytropic gas dynamics; in addition, u(x,t) is assumed to respect
the usual entropy conditions connected with the conservative Euler
equations. We show that such entropy solutions of the gas dynamics equations
satisfy a minimum entropy principle , namely, that the spatial minimum of their
specific entropy. Ess lnf_s( u( x, t ) ) , is an increasing function of time. This

X

principle equally applies to discrete approximations of the Euler equations
such as the Godunov-type and Lax-Frledrlchs schemes. Our derivation of this
minimum principle makes use of the fact that there is a family of generalized
entropy functions connected with the conservative Euler equations.

Research was supported in part by NASA Contract No. NASl-17070 while the
author was in residence at ICASE, NASA Langley Research Center, Hampton, VA
23665-5225. Additional support was provided in part by NSF Grant No. DMS85-
03294 and ARO Grant No. DAAG29-85-K-0190 while in residence at the University
of California, Los Angeles, CA 90024.

Bat-Sheva Foundation Fellow

100

1. INTRODUCTION

Many phenomena in continuum mechanics are modeled by hyperbolic systems of
conservation laws

|f+ I ^^—=0> (x = (x^,...,x^),t)eR><[0,<»), (1.1)

k=l k

where

.(k) _ ^(k), ., _ fAk) _. ^(k)^T

f ' = f^ '(u) = [f^ ,»»»,f^ ) are smooth nonlinear flux mappings
of the N-vector of conservative variables u = u(x,t) = (u. ,»««,u ) .
Friedrichs and Lax [3] have observed that the hyperbolic nature of such models
is revealed by the property of most of those systems being endowed with a
generalized

Entropy Function ; A smooth convex mapping U(u) augumented with entropy flux
mappings F = F(u) = (F (u),'»«,F^ (u)), such that the following
compatibility relations hold

uT ^(k) ^ p(k)T k=l,2,...,d. (1.2)

u u u

T
Multiplying (1.1) by U and employing (1.2), one arrives at an equivalent

formulation of the compatibility relations (1.2), namely, that under the

smooth regime we have on top of (1.1) the additional conservation of entropy

I^.I^.O. (1.3)

'' k-l »\

(k)
Owing to the nonllnearlty of the fluxes f (a), solutions of (1.1) may

develop singularities at a finite time after which one must admit weak

101

solutions, i.e., those derived directly from the underlying integral
conservative equations. Considering (1.1) as a strong limit of the
regularized problem,

., d (k) d .2

37+ I -TZ ul —7. U + 0, (1.4)

^^ k=l ^\ k=l 9x,2 V

k

then following Lax [9] and Krushkov [8], we postulate as an admissibility
criterion for such limit solutions an entropy stability condition which
manifests itself in terms of an

Entropy Inequality ; We have, in the sense of distributions.

Weak solutions of (1.1), which in addition satisfy the inequality (1.5)
for all entropy pairs (U,f) connected with that system, are called entropy
solutions .^ ^^ Having a (weakly) nonpositive quantity on the L.H.S. of (1.5)
is thus a consequence of viewing these entropy solutions as limits of
vanishing dissipativity mechanisms. In particular, the inequality (1.5)
implies that the total entropy in the domain decreases in time (we assume
entropy outflux through the boundaries)

^/_ U(u(x,t))dx < 0. (1.6)

X

•'Krushkov [8, p. 241] has termed such solutions simply as generalized
solutions.

102

In this paper, we consider entropy solutions.

u = (p.B.E)'^ (1.7a)

of the Euler equations. These equations govern the inviscid polytropic gas

dynamics, asserting the conservation of the density p, the momentum

T m

in = (m. ,m„,m„) , and the energy E. Let q = — denote the velocity field of

such motion. Then, expressed in terms of the pressure, p,

P = (y-1)*[E -V2*p|q| ]. Y = adiabatic exponent, (1.7b)

(2)
the corresponding fluxes in this case are given by^ '

f^*"^ = (m^,qj^.m+ p.e^''\qj^(E + p))"^, k= 1,2,3. (1.7c)

The main result of this paper asserts that entropy solutions of Euler
equations satisfy the following

Minlmtim Principle: Let m = u(x,t) be an entropy solution of the gas
dynamics equations (1.7) and let

S(x,t) = S(u(x,t)) = ln(pp"^) (1.8)

^^^With e^^^ denoting the unit Cartesian vectors e^^^ = 6j^ . .

103

denote the specific entropy of such solution. Then the following estim ate
holds

Ess inf S(x,t) > Ess inf S(x,t = 0). (1.9)

|x|<R |x|<R+fq

Here qj^^^^ stands for the maximal speed |q| in the domain .

The proof of this assertion is provided in Section 3 below. Prior to that
we elaborate in Section 2 on the entropy inequality connected with the gas
dynamics equations. In particular, Harten [5] has shown that there exists a
whole family of entropy pairs associated with these equations, a fact which is
essential in our derivation of the minimum principle.

As an immediate consequence of the minimum principle, we conclude that
Ess inf_S(x,t) is an increasing function of t for every entropy solution of

X

(1.7). The following argument sheds additional light on this conclusion in
the case of a piecewise-smooth flow. To this end, an arbitrary particle
currently located at (x,t) is traced backwards in time into its initial
position at t = 0. Since the specific entropy of such particle remains
constant along the particle path — except for its decrease when crossing
backwards shock waves, it follows that its value S(x,t) is greater or equal
than that of the initial spatial minimum Ess inf S(x,t = 0), as asserted.

X

In contrast to the above 'Lagrangian' argument, the derivation of the minimum
principle outlined below, is purely an 'Eulerian' one. It enables us to relax
the regularity assumption on the flow, and — since we do not follow the
characteristics, it equally applies to discrete approximations of the Euler
equations.

104

In Section 4 we consider approximate solutions of the Euler
equations, w(x ,t), which respect the entropy decrease estimate (1.6),

y U(w(x ,t + At))Ax < y U(w(x ,t))Ax . (1.10)

We note that such approximate solutions are obtained by entropy stable
schemes satisfying the cell entropy inequality

U(w(x^,t +At)) <U(w(x^,t)) + I 4r-[F^+V -F^^VJ» a. 11)

k=l Ax ^ ^

V

e.g., the Godunov-type and Lax-Friedrichs schemes [6]. We have

Minimum Principle: Let w(x ,t) be an approximate solution of the gas
dynamics equations (1.8) and let

S(x^,t) = S(w(x^,t)) = In(pp^) (1.12)

denote the specific entropy of such solution. Assume that its total entropy
decreases in time, (1.10). Then the following estimate holds

S(x,t + At) > Min[S(x ,t)]. (1.13)

V

In the case of entropy stable schemes, (1.11), a more precise estimate is
obtained which takes into account the support of the schemes' stencil.

The inequality (1.13) leads to an a'priori pointwise estimate on the
approximate solution w(x,t). Such pointwise estimates play an essential role

105

with regard to question of the convergence of entropy stable schemes. In
particular, DiPerna [2, Section 7] has recently shown that in certain cases,
such (two-sided) estimates are sufficient in order to guarantee the
convergence of such schemes.

2. GENERALIZED ENTROPY FUNCTIONS OF THE EULER EQUATIONS

We consider the Euler equations for polytropic gas

3_
8t

+ I i—
11 9x,
k=l k

m,

(k)

qj^m + pe
q^(E + p)

= 0.

(2.1)

It is well-known, e.g., [1], that for all smooth solutions of (2.1) the
specific entropy'^''

S(x,t) = ln(pp"^),

remains constant along streamlines, i.e..

DS 9S . ^ 3S „

(2.2a)

Let h(S) be an arbitrary smooth function of S. Multiplying (2.2a) by
ph'(S) — prime denoting S-dif ferentiation, we find

(3)Aft

er normalization, taking the specific heat constant to be c^ = 1,

106

8h(S) . ? 9h(S)

k=i -^ ^""k

= 0.

Adding this to the continuity equation which is premultiplied by h(S),

3 am
If h(S) + I r^MS) = 0, (2.2b)

we obtain after changing sign, a conservative entropy equation like (1.3)
which reads [5]

3

1^ [-ph(S)] + I Ip [-m^MS)] = 0. (2.3)

k=l k

In order to comply with the further requirement of being a generalized entropy

function, U(u) = -ph(S) has to be a convex function of the conservative

T
variables u = (p ,«,E) . A straightforward computation carried out by Harten

[5, Section 2] in the two-dimensional case shows that the Hessian U is

uu

positive definite if and only if

p[h'(S) - Y-h"(S)] > 0.

Excluding negative densities we may summarize that there exists a family of
(generalized) entropy pairs (U,F) associated with Euler equations (2.1),

U(u) = -ph(S), F^^^(u) = -mj^h(S) k = 1,2,3, (2.4a)

generated by the smooth increasing functions h(S) which satisfy

107

h'(S) - T'h"(S) > 0. (2.4b)

3. A MINIMUM ENTROPY PRINCIPLE

T
Let u = (p,iB,E) be an entropy solution of the gas dynamics equations

(2,1). Such a solution is characterized by the entropy inequality (1.7)

'-^*l'-^<0 C3.0

^^ k=l ^\ -

which holds for all entropy pairs (U,l) connected with the equations.

To derive a minimum principle, we shall make use of an argument due to Lax
[9, Section 3]. We begin with

Lemma 3.1: Let u be an entropy solution of the gas dynamics equations
(2.1). Then for all nonpositive smooth increasing functions h(S) satisfying
(3.2b), we have

/ p(x,t)«h(S(x,t))dx > / p(x,0).h(S(x,0))dx. (3.3)

|x|<R |xl<R+fq

- ' '= ^max

Here qj^^^^ denotes the maximal speed |q| in the domain .

Proof : As in [10, Theorem 4.1] we integrate the entropy inequality (3.2a)

over the truncated cone C = { Ixl < R + (t - T).q lO < t < t} ; if we let

' ' ' = ^max ' = = J »

(n ,n) denote the unit outward normal, then by Green's theorem

108

/ ph(S).

dc

k=l

3x > 0.

(3.4)

The Integrals over the top and bottom surfaces give us the difference between

the left and right-hand sides In (3.3) and by (3.4) this difference is bounded

from below by

d

9x.

-/ ph(S).
mantle

"o ■" J^ ^k^k

The result follows upon showing that the last quantity is nonnegative .
Indeed, since by assumption -ph(S) > 0, this is the same thing as

k=l

on the mantle we have

(nQ,n) =

ATT"

max

Vx' |-|

and hence

Hn + I qk"k =

k=i "" /TT^

r*max ^

max

^=' 1^1/ "ATT

max

- I

3 |q,

>

k=l

'max

as asserted.

The discussion in Lemma 3.1 was restricted to smooth function h(S); by

passing to the limit, its conclusion (3.3) follows for any nonpositive

nondecreasing function h(S) satisfying (3.2b), whether smooth or not.

109

To derive the minimum entropy principle, we now make a special choice of
such function, h(S), given by

h(S) = Mln[S - Sq.O], Sq = Ess inf S(x,0). (3.5)

lx|<R+t.q
' ' = max

The nonposltlve function h(S) is a nondecreasing concave one, hence
admissible by (3.2b), and consequently (3.3) applies

/ p(x,t)'Min[S(x,t) - SQ,0]dx >
' '- (3.6)

/ p(x,0).Min[S(x,0) - SQ,0]dx.

lxl<R+fq
' '— max

Now, by the choice of Sg, the integral on the right of (3.6) vanishes since

Min[S(x,0)-SQ,0] does. The inequality (3.6) then tells us that the integral

on the left is also nonnegative. But since the integrand on the left is by

definition nonpositive, this can be the case provided this integrand vanishes

almost everywhere; that is, we have for almost all x, |x| < R

S(x,t) > Sq = Ess inf S(x,t=0)

|x|<R+fq
' - max

and (1.9) follows.

The minimum entropy principle was deduced from the entropy inequality
(3.2), which in turn was postulated based on the formal regularization
introduced in (1.4). In general, other regularizations equally apply; In

110

particular, Euler equations are usually sought as the vanishing viscosity
limit of the Navler-Stokes equations (here we take for simplicity the one-
dlmenslonal case)^ ■'

3_

at

[■p]

m

m

^k

qm + p

_E_

Lq(E + p)

r

= y

9x

dq
9x

ia

9x

y 4- 0.

(3.7)

Do the (generalized) entropy Inequalities (3.2) remain valid on the basis of
such limit? To answer this question we first note that If U(u) Is any
entropy function, then thanks to Its convexity the mapping u -»■ v = U Is
one-to-one, and hence one can make the change of variables u = u(v). Harten
[5] has shown that such change of variables by each member of the family of
entropy functions (2.4) puts the viscosity terms on the right of (3.7) Into a
negative semldeflnlte form. This makes apparent the dlsslpatlve effect of
these viscosity terms. Indeed, If T = c • E -V2*|q| denotes the absolute
temperature, then direct manipulation of (3.7) yields, e.g., [1, Section 63],
[12, Section 6.10] ,

Iy [ph(S)] + 1^ [mh(S)] = p.h(S) ^

(3.8)

from which we recover the entropy Inequality (3.2a) for all smooth Increasing
functions h(S). We note that the convexity condition was not assumed In this

^ •'with y combining the two viscosity coefficients In the general Navler-
Stokes equations.

Hi

case. The merit of using the convexity condition, however, is that it enables
us to deal with more general artificial viscosity terms, other than those
appearing in the Navier-Stokes equations. Such artificial viscosity terms are
frequently encountered in finite-difference approximations to the Euler
equations; a specific example of this kind is studied in the next section.

Finally we would like to remark on the previously mentioned Navier-Stokes
equations. Our discussion above took into account only the viscosity
contribution, neglecting heat conduction. Hughes, et al., [7] have shown that
when the heat flux is also added, compare (3.7),

3_
9t

8x

m
qm + p
q(E + p)

= y

- -

3

ax

3q
3x

3q

+ K

3
3x

3T
3x

(3.9)

with K denoting the heat conductivity constant, then only the 'physical'
entropy, U(u) = -pS survives as the one which puts the additional heat flux
Into a symmetric negative-definite form. We would like to note in this
connection the difference limit behavior of the Navier-Stokes flows depending
on the viscosity and heat conductivity; Gilbarg [4] has shown that as < -»■
keeping y fixed, we are led to a continuous thermally nonconducting shock
layer, whereas for y -»■ with k fixed the convergence is to a (generally)
discontinuous nonviscous shock layer. Consequently, the viscosity rather than
the heat flux should play the major rule in an appropriate regularization
model for the Euler equations.

112

4. DISCRETE APPROXIMATIONS OF THE EDLER EQUATIONS

In this section we consider approximate solutions of the Euler
equations, w(x ,t), whose total entropy decreases in time, compare (1.10)

Iv"^*^\''' "^ At))Ax^ < 5:^U(w(Xy,t))A7^. (4.1)

Estimate (4.1) holds for all entropy functions U = -ph(s) in (2.4). By
passing to the limit, this applies to our previous choice of the function
h(s) in (3.5)

h(s) = Min[S - 8^,0], (4.2a)

this time with a constant Sg which is taken to be

Sq = Min S(w(x ,t)). (4.2b)

V

By our choice of Sq, we have U(w(x ,t)) = 0. The inequality (4.1) tells us
that the left-hand side is therefore, nonnegative; consequently

S(x,t + At) - Sq > h(S(x,t + At)) >

and (1.13) follows.

Approximate solutions which fulfill the required estimate (4.1) can be
obtained by entropy stable schemes satisfying the cell entropy inequality
(1.11)

U(w(x^,t + At)) < U(w(x^,t)) + I 4- [F^!j\, - F^^\ ]. (4.3)

k=l Ax ^ '•

113

Examples of such entropy stable schemes include the Godunov-type and Lax-
Friedrichs schemes, e.g., [6]. A more precise minimum principle follows in
these cases, taking into account the support of the schemes' stencil. In
particular, the (one-dimensional) Godunov scheme results from averaging of two
neighboring Riemann problems [6], each of which satisfies (1.9). Consequently
we have the

Minimum Principle (of the Godunov scheme): Let w(x ,t) the Godunov

approximate solution to the Euler equations (2.1). Assume that the

appropriate CFL condition is met. Then the following estimate holds

S[w(x ,t + At)) > Min S(w(x ,t)). (4.4)

^ v-l<jj<v+l ^

Since the Lax-Friedrichs scheme coincides with a staggered Godunov's solver,
the same conclusion, (4.4), holds. Another way to see this is outlined below;
it makes no reference to Rlemann's solution and can be generalized to the
multidimensional problem.

To this end, we approximate the (for simplicity — one-dimensional) Euler
equations with the Lax-Friedrichs scheme

w(x^,t + At) = 1 [w(x^^^,t) + w(x^_j,t)]

(4.5)
-H'^"^^+l''^^ -f(w(x^_l.t))], X e||.

We remark that the Lax-Friedrichs scheme can be derived from center
differencing of the regularization model (1.4) . Lax has shown [9, Theorem

114

1.2] that if X

At

is sufficiently small, then solutions of this difference
scheme satisfy the following cell entropy inequality

U(w(x^,t + At)) ^

U(w(x^^j,t)) + U(w(x^_^,t))

(4.6)

- I [F(w(x^^^,t)) - F(w(x^_^.t))]

for aU entropy pairs (U,F) = (-ph(S) ,-mh(S)) in (2.4). by passing to the
limit, this applies to our previous choice of the function h(S) in (3.5)

h(S) = Min[S - S„,0],

(4.7a)

this time, with a contant Sq which is taken to be

Sq = Min[s(x^^^,t), S(x^_j,t)]

(4.7b)

The inequality (4.6) now reads

p(x^,t + At).h(s(x^,t + At)) I

1 + Xq(x^_^,t)

2 P(%-i.t)-h(S(x^_^,t))

+ r-^— P(x ,^, .t).h(S(x ,,, ,t))

(4.8)

v+r

By our choice of the funtion h(S) in (4.7), we have h(S(x .,t)) = 0. The
inequality (4.8) tells us that the left-hand side is therefore nonnegative;
consequently

115

< h(S(x ,t + At)) < S(x,t + At) - Sq

and the following minimum principle follows

S(w(x ,t + At)) > Min S(w(x .1 .t)),

116

REFERENCES

[1] R. Courant and K, 0. Friedrichs, Supersonic Flow and Shock Waves ,
Interscience, New York, 1948.

[2] R. J. DlPerna, "Convergence of approximate solutions to conservation
laws," Arch. Rational Mech. Anal ., Vol. 82 (1983), pp. 27-70.

[3] K. 0. Friedrichs and P. D. Lax, "Systems of conservation laws with a
convex extension," Proc. Nat. Acad. Sci. U.S.A ., Vol. 68 (1971), pp.
1686-1688.

[4] D. Gilbarg, "The existence and limit behavior of the one-dimensional
shock layer," Amer. J. Math .. Vol. 73 (1951), pp. 256-274.

[5] A. Harten, "On the symmetric form of systems of conservation laws with
entropy," J. Comput. Phys ., Vol. 49 (1983), pp. 151-164.

[6] A. Harten, P. D. Lax, and B. Van Leer, "On upstream differencing and
Godunov-type schemes for hyperbolic conservation laws," SIAM Rev ., Vol.
25 (1983), pp. 35-61.

[7] T. J. R. Hughes, L. P. Franca, and M. Mallet, "Symmetric forms of the
compressible Euler and Navier-Stokes equations and the second law of
thermodynamics," Comput. Methods Appl. Mech. Engrg ., to appear.

117

[8] S. N. Krushkov, "First-order quasilinear equations in several
independent variables," Math. USSR-Sb ., Vol. 10 (1970), pp. 217-243.

[9] P. D. Lax, "Shock waves and entropy" in Contributions to Nonlinear
Functional Analysis (E. H. Zarantonello, ed.), pp. 603-634, 1971.

[10] E. Tadmor, "Skew-self ad joint form for systems of conservations laws," J.
Math. Anal. Appl ., Vol. 703 (1984), pp. 428-442.

[11] E. Tadmor, "The numerical viscosity of entropy stable schemes for
systems of conservation laws. I.," NASA Langley Research Center, ICASE
Report 85-51, NASA CR-178021, 1985.

[12] G. Whitham, Linear and Nonlinear Waves, Wiley-Interscience, 1974.

118

A SPECTRAL MDLTIDOMAIN METHOD
FOR THE SOLUTION OF HYPERBOLIC SYSTEMS

David A. Kopriva

Florida State University

and

Institute for Computer Applications in Science and Engineering

ABSTRACT

A multidomain Chebyshev spectral collocation method for solving hyperbolic
partial differential equations has been developed. Though spectral methods
are global methods, an attractive idea is to break a computational domain into
several subdomains, and a way to handle the interfaces is described. The
multidomain approach offers advantages over the use of a single Chebyshev
grid. It allows complex geometries to be covered, and local refinement can be
used to resolve important features. For steady-state problems it reduces the
stiffness associated with the use of explicit time integration as a relaxation
scheme. Furthermore, the proposed method remains spectrally accurate.
Results showing performance of the method on one- dimensional linear models
and one- and two-dimensional nonlinear gas-dynamics problems are presented.

Research was supported by the National Aeronautics and Space Administration
under NASA Contract Nos. NASl-17070 and NASl-18107 while the author was in
residence at ICASE, NASA Langley Research Center, Hampton, VA 23665-5225.

119

1. INTRODUCTION

In this paper we address the problem of efficiently computing Chebyshev
spectral collocation approximations to quasilinear hyperbolic systems of the
form

Qj. + A(Q)Q^ + B(Q)Qy = x,y DCR^, t > (1)

with appropriate boundary and initial conditions. Here, Q is an m-vector
and A and B are mxm matrices. This system is hyperbolic if for any
constants k^ and k2 the matrix T = k, A + k„ B has only real eigenvalues
and there exists a similarity transformation matrix, P, such that FTP = A
is a real diagonal matrix.

In particular, we are interested in the solution of the Euler equations of
gas dynamics which form a system of this type. The use of the nonconservation
form is justified for problems in which shocks are fitted and in this
situation spectral methods work well [1]. Problems of the type presented in
Ref. [1] provide the motivation for what follows.

The typical Chebyshev spectral collocation procedure for the solution of
the system (1) is described in several reviews such as those of Gottlieb,
Hussainl, and Orszag [2], and Hussaini, Salas, and Zang [3]. First, the
domain of interest is mapped onto the square D' = [-1 ,l]x[-l , 1] and an
(N+M) X (M+1) point mesh is generated with the collocation points defined by

x^ = - cos(iTT/N) i = 0,1, •••,N

y = - cos(JTr/M) j = 0,1,...,M. (2)

120

Mesh point values of Q, designated by Q-f^> are associated with each of the

collocation points (xj^.y. ). A global Chebyshev interpolant of order N in

the X direction and order M in the y direction is then put through the
mesh point values

N,M

n,m=0

Approximations to the derivatives at the collocation points are computed
by differentiating the interpolant and evaluating the resulting polynomial at
the collocation points. The computation of the derivatives can be
accomplished in one of two ways (see Gottlieb, et al., [2]): The first is to
take advantage of the fact that the sums for both the interpolant and its
derivative reduce to cosine sums at the chosen collocation points. For
example

dQ N,M N,M

- = I ^n T:(x)T„(y) = I b_T (x)T„(y) (4)

where

dx ^ _ nm n m " " _, nm n " ' m
n,m=U n,m=0

Nm '

b„ . = 2Na (5)

N-l,m nm ^ '

and

^r,K^ = K^o ^ + 2(n + Da . , for < n < N - 2.
n nm n+z,m n+l,m — — •

The constant c^ is defined as c^^ = 2 for n = 0,N and c = 1
otherwise. The advantage of this form is that a fast cosine transform can
compute the derivatives along each y line in 0(N log N) operations.

121

The other approach to computing the derivatives is to write the
differentiation operation as the product of a differentiation matrix and the a
vector of the Qij's. For example, along each y line the x derivative is

dQ

^d^)j = D(Qp)j (6)

r * T

where (0^)^ = [Qq^^ Q^^j ••"• %^^] and the elements of the matrix D are
defined in Gottlieb et al., [2]. The amount of work with this procedure is
of 0(N ). What one loses in efficiency one gains as flexibility in the
number of mesh points that can be used in each direction without adding
storage.

No matter which way the spatial derivatives are computed, it is important
to note that computing the Chebyshev derivative approximations requires only
mesh point values. Derivatives at the end points require only points interior
to the mesh so no extra procedure is required to compute derivatives at
boundaries.

Once the spatial derivatives are approximated, what results is a system of
ordinary differential equations in time for the variation of the solution at
each collocation point (Method of Lines). Because the differentiation matrix
is full, explicit methods are typically used to integrate the semi-discrete
equations. In this paper, all time integrations will be performed with a
fourth-order Runge-Kutta method.

The advantage of using this spectral method to solve (1) is that for

CO

solutions which are C (D) , the accuracy is better than any polynomial order
(Canuto and Quarteroni, [4]). This is usually called "spectral accuracy" and
asymptotic behavior can be observed if there are enough grid points to

122

adequately resolve the solution. It is thus possible to compute to a given
spatial accuracy with fewer grid points than required by typical low-order
finite difference approximations.

Balancing the high accuracy of the spectral method, however, are some
major disadvantages of the typical Chebyshev collocation approach:

(1) It may not be easy or even possible to map D ->• D' globally.

(2) The collocation point distribution is global and predetermined. Local
refinement of the mesh is not possible.

(3) The points are concentrated near the boundaries where they are
typically not needed for hyperbolic problems.

(4) If explicit time integration is used the time step restriction in one

2
dimension is proportional to 1/N .

(5) For complete flexibility in the number of mesh points which can be
used, the derivatives cost of 0(N ) in each direction.

These problems can be reduced significantly by breaking up the region D
into several subdomains \ each of which has its own Chebyshev grid. With a
stable and efficient method for computing the interfaces, the advantages of
such an approach would be:

(1) Complicated geometries can be covered.

(2) Points can be distributed with some flexibility; local refinement is
possible.

(3) In one dimension, with N points and K subdomains, the time step

2
restriction increases to At « K/N .

(4) Derivative evaluation work with matrix multiplication decreases to
K(N/K)^ or 1/K that of a single grid.

123

The idea of breaking up the computational domain into subdomains each with
a different grid is not new. For finite difference methods this is a
currently popular approach (e.g., [5]). For spectral methods, however,
previous applications have been limited to elliptic and parabolic problems.
Orszag [6] first applied such a technique to solve elliptic problems. He
enforced continuity of the function and its first derivative as the interface
condition. Metivet and Morchoisne [7] and later, Morchoisne [8] computed
multidomain solutions to the Navier-Stokes equations. Recently, Patera [9]
and Korczak and Patera [10] have been using a spectral element method to solve
the incompressible Navier-Stokes equations. Their method is very similar to
the p finite-element methods developed by Babuska (see [10]) but uses
Chebyshev interpolants. The treatment of the convective terms, however, does
not lend itself to purely convective problems. For these problems, we
describe the method below.

2. MULTIDOMAIN APPROACH

In this paper, we will break up the physical domain, D, into K
subdomains Dj^ which do not overlap except for the common boundary points.
Figure 1 shows a rectangular two-dimensional example of the situation with
four subdomains. Each of the D^ are mapped onto a square [-1 ,1 ]x [-1 , 1] .
Spatial approximations at interior points of each subdomain are computed in
the usual way. Across an interface, however, there are two values of the
normal derivative. For example, at the y coordinate line interface between
D, and D2 in Figure 1 , derivative approximations are available from the
left and from the right. The problem is to choose properly information from

124

the right and the left to give a stable and consistent approximation to the
differential equation at the interface.

Before discussing a multidomain method for the boundary value problem (1),
we will first examine the one-dimensional case. In one dimension, we seek
interface algorithms of the semidiscrete form

|i + AL|Q!: + AR|Q!=o (7)

9t 8x 3x

where Q denotes the value of Q at an interface and the derivatives
superscripted with L and R denote the two spectral approximations computed
in the left and right, respectively. For consistency, we require that

A^ + A^ = A (8)

and for efficiency we want A^ and A^ to be computed with little more work

than is required for the computation of A itself.

To generate the coefficient matrices, consider first the linear scalar

hyperbolic equation

u, + Xu = X > 0. (9)

t X

Because the equation is hyperbolic, it is clear that the common interface
point should depend only on information propagated from the left. Thus, the
approximation should be

|^+x|^=o. (10)

9t 8x

125

This is, of course, just upwind differencing at the interface and is
equivalent to the way Gottlieb and Orszag [11] handled a tau approximation to
equation (9). To simplify the computational logic to include cases where the
coefficient, A, is of either sign, the approximation (10) can be written as

#-V2(X.|x|)|^.l/2(X-|x|)|^=0.

(11)

If we now consider that this equation is a single component of a
diagonalized system, where the diagonal matrix

A =

1

we can write the system as

n

= P ^ AP,

|^+V2(Af |A|)|5-+1/2(A- |A|)|5-=0

(12)

where |a| = p|a|p . Formally, this is nothing more than the method of
characteristics in one dimension.

We now propose to avoid the computation of the matrix absolute value by
approximating it with a diagonal matrix

* -1 *

II ** — I «
a| « px IP ^ = X I

(13)

where X is chosen to lie between the largest and smallest elements of |a|.
The boundary scheme is now of the form of Eq. (7) with

126

A^ = 1/2 (A + X* I) A^ = 1/2 (A - X I). (14)

This choice of coefficient matrices always has proper upwind dominance on

all of the characteristic variables, but includes some downwind influence. To

see this, re-diagonalize the system (7) and use u as the n component of

the diagonalized system. Then the approximation to the method of

characteristics causes the characteristic variables at the interface to be
approximated by

#-V,(x„.x*]|^.iMx„-x*)|^.o. (15)

In fact, this can be viewed as the purely upwind scheme with an error term:

For the X > case,
n

l^+X |ii= (X*-X )(|ii^-|iL:). (16)

9t n 3x ^ n-'^9x 9x -'

Thus, we have the spectrally accurate upwind approximation with an error
term proportional to the difference of the right and left spectral
derivatives. If the solution has the necessary smoothness, this difference
should also decay spectrally and spectral accuracy of the approximation should
be retained.

We will study the stability of the multidomain method with the interface
approximation (14) numerically. An analytic study of stability is not
possible at this time. Stability theory for Chebyshev approximations to
hyperbolic initial-boundary value problems is not advanced enough to analyze
an approximation which introduces some downwind influence at the interface.

127

We consider the two-domain approximation of the scalar equation (9) with
the interface approximation (12) with X = 1. The line segment [-2,2] is
divided equally into two domains of [-2,0] on the left and [0,2] on the right.
The semidiscrete approximation can be written as a system of ordinary
equations with the two-domain coefficient matrix

.R

(17)

L R
where D and D are the single domain differentiation matrices for the

left and the right, modified to include the interface approximation. For this

system to be time stable, that is, the solution does not grow unboundedly as

t ->- ", the eigenvalues of the coefficient matrix must have negative real

parts.

Figure 2 shows how the eigenvalues change as A varies when 6 points are

*
used. The case of A = corresponds to simple averaging and is clearly not

*
time stable. Choosing A > large enough moves the eigenvalues into the

left half of the complex plane and the resulting approximation is time

*
stable. The case of A = 1 is the purely upwind case and the eigenvalues

decouple into two single-domain patterns. If A is chosen equal to, or

larger than, the wave speed, A , the approximation has the effect of adding a

purely dissipative term to the equation and two purely real eigenvalues are

*
created. If A is very much larger than A , however, the eigenvalues

migrate to the right of the imaginary axis. The range of A's for which the

approximation is stable decreases as the disparity in the number of points

128

becomes larger; for very stiff systems, it may be necessary to use \a\

*
instead of X at the interface.

It is interesting to note that the reverse situation, where there is more
resolution on the upstream side of the interface, does not show this behavior
and is stable for all X ^ ^' ^°^ systems, this means that X should be
chosen to be only slightly larger than the smallest eigenvalue representing a
characteristic moving from the coarse to the fine grid. For systems, this
means that X should be chosen to be only slightly larger than the smallest
eigenvalue representing a characteristic moving from the coarse to the fine
grid. We note, however, that the examples on which the scheme has been tested
show that the approximation is robust over a wide range of choices of X .

In two dimensions, the upwind weighted approximation is used in the
direction perpendicular to the interface. Returning to Figure 1, along x
coordinate lines, the y derivatives are continuous across the Interfaces
except at corners. At points not on the corners, then, we propose using

|Qi..A^9i+A^|Q!+B|i=0 (18)

3t 8x 9x 8y

where A^ and A^ are defined as above. Along x coordinate interfaces,

|Qi+A|i-.B^|s!:+B^|Q!=0 (19)

dt 3x 9y 8y

where b''' = V2 (B + p* I) and B^ = V2 ( B - y* I) and p is an approxima-
tion to the eigenvalues of B. At corners, the weighted approximations are
used in both directions.

129

3. NUMERICAL EXAMPLES

Numerical experiments on four model problems in one and two dimensions
will be presented. The models include the scalar one-dimensional hyperbolic
initial boundary value problem for a travelling Gaussian pulse, a linear
system in one dimension, quasi-one-dimensional flow in a converging-diverging
nozzle, and the transonic Ringleb problem. The Ringleb flow models the smooth
nonlinear transonic flow in a curved duct and has an exact solution to which
to compare.

A. Solution of a Linear Scalar Problem

The solution to the linear scalar problem

9u , „ 3u

TF"*" 2 3ir= ° xe[-2,2], t > (20)

u(x,0) = exp(-(x - Xq)^/0.3) xe[-2,2]

u(-2,t) = exp(-(x - t - Xq)^/0.3) t >

can be used to examine the effects of varying X in the spatial

approximation described in Eq. (15). The time integration for this and all

following examples was a fourth-order Runge-Kutta technique. For this and the

next model problem the time step was chosen so that the temporal errors were

on the order of 10 . The main questions to be answered here are the effect

*
of the X ^ 2 on the accuracy of the solution and if reflections are a

problem at the interface. Figure 3 shows the computed (circles) and exact

(line) solutions for the pulse after it has propagated through the interface

at x = for two distributions of the mesh points and X = 6.

130

The interface approximation Eq. (15) degrades the accuracy of the solution

*
when compared to the purely characteristic interface, X = 2, if equal

resolution is not provided in each subdomain. In no case, however, is the

global Ln error larger than the global error for the characteristic inter-

face. Furthermore, if A remains fixed and the total number of points is

increased, the error decay remains spectral. Figure 4 shows the pointwise

errors of the solution to Eq. (20) for the situations represented in Figure 3

*
as X is increased beyond the characteristic value of 2. The situation is

worse when more resolution is used upstream of the interface because the

approximation includes more and more downwind influence as X is

increased. In a practical computation, the effect of the boundary

approximation would not be important if the solution were equally resolved in

all subdomains.

Reflections at the interface are not visible in Figure 3 even though there

is a factor of two difference in the number of collocation points. Gottlieb

and Orszag [11] also noticed this for a tau approximation to the scalar wave

equation. This is typical for the spectral approximations; examples with up

to a factor of three and four in the ratio of the number of mesh points have

not shown spurious reflections off of the interface.

B. A Linear System Example

The accuracy of the interface approximation will now be demonstrated with
the 2x2 linear system

u'

+

"1

2"

u'

V

t

L2

1

V

-"x

X e[-2,2], t > 0.

(21)

131

The coefficent matrix has eigenvalues +3 and -1 so the system has
information which propagates in both directions and with different speeds
across the interface at x = 0. The initial and boundary conditions were
chosen so that the characteristic variables were the Gaussian pulses used in
the scalar problem, Eq. (20). The coefficient X for this case was chosen
to be the maximum eigenvalue, X = +3. Figure 5 shows the results for the two
components of this system at a time when the characteristic pulses have
crossed the Interface. In Figure 5a there are twice as many points to the
left of the Interface as to the right and this is reversed for Figure 5b. The
symbols represent the computed solutions and the solid lines represent the
exact solutions.

A study of discrete L2 errors for the system computations Is shown in
Tables I through III. Clearly, the error is spectral for all three
situations. In fact, for an equal number of mesh points on either side of the
interface, the error decay is exponential. For the problem of propagating
pulses, where the features needing higher resolution are continually moving,
it is not surprising that the best errors are obtained when there are an equal
number of mesh points on both sides of the Interface.

C. Quasl-One-dlmensional Nozzle Flow

One potential point of concern in using the interface approximation given
by Eq. (14) regards the stability of cases where one of the eigenvalues of the
coefficient matrix is much larger than any other. Such a situation occurs at
sonic points in an ideal gas flow where one of the characteristic speeds
actually vanishes.

132

To test this situation the nonlinear problem of steady gas flow in a
quasi-one-dimensional converging-diverging nozzle was solved with the
multidomain method where an interface was placed at the sonic point. The
quasilinear form of the Euler gas dynamics equations for time-dependent flow
in a quasi-one-dimensional nozzle without shocks can be written as

■p'

+

u

y"

■p'

_,

■yuA^(x)/A(x)'

u

t

a^/Y

u

u

X

(22)

where P is the logarithm of the pressure, u is the gas velocity, y is the
ratio of specific heats, and a is the sound speed. The coefficient matrix
has eigenvalues of u + a and u - a so that one of them is zero at a sonic
point. The steady flow is found as the large time limit of the unsteady flow
described by (22).

The nozzle area is given by A(x) = x/2 + 1/x so the throat occurs at
X = /2. For the cases run, a subsonic inflow boundary was placed at x = 0.2
and characteristic boundary conditions were used. After the gas accelerates
through the sonic value at the throat, it leaves the nozzle supersonically so
no boundary conditions are applied at the outflow.

For the gas dynamics calculations in one dimension, X =V2(|u+a| + |u-a|)
was chosen since this corresponds to the diagonal elements of the absolute
value of the coefficient matrix. Although the problem was solved for domain
interfaces in both the subsonic and supersonic portions of the nozzle, only
results for a single Interface at the sonic point will be shown here. (The
two-dimensional example below will include a variety of interface placements.)

133

Figure 6 shows the steady pressure in the nozzle computed with two domains
and twice as many mesh points on the right as on the left. Our tests on a
variety of grids have not shown any stability difficulties In computing steady
flows when placing the interface at a sonic point.

D. Two-Dlmensional Transonic Flow

A more complicated problem Is the two-dimensional transonic Ringleb
flow. This problem allows us to study the computational efficiency of the
multldomain solution algorithm as outlined in the Introduction. Kopriva, et
al., [12] used this problem for a comparison of the performance of the
spectral method with a second-order finite-difference method. In this section
we will compare the multldomain spectral method with the single domain
spectral method.

The Ringleb flow is a simple example of a two-dimensional transonic flow
for which there is an exact solution. (See, for example, Courant and
Friedrichs [13].) The streamlines of the physical space solution appear at
large distances as parabolas which are determined from a special hodograph
solution of the potential equation for steady irrotational Isentroplc flow.
By choosing two streamlines to represent solid walls, this problem models a
steady transonic flow in a duct. Figure 7 shows the Mach contours of one such
duct flow.

Again we will look for the large time solution of the unsteady gas
dynamics equations, this time in two dimensions. The problem in the curved
duct shown in Figure 7 is mapped onto a rectangle in the stream function-
potential (i|),(j)) coordinate system derived from the exact solution. In this

134

coordinate system, the unsteady equations can be written as

Q. = -R

(23)

where R is the steady state residual

R = AQ^ + %•

(24)

Since the solution is irrotational, the solution vector is chosen to be

Q = [P u v]

(25)

and the coefficient matrices are

A =

U

<j) d)
X y

a (b /y U
^x

a d) /y U

y

U

B =

lb iL
X y

a J) /y V
^x

a t /y

y

As before, P represents the logarithm of the pressure and (u,v) represent

the velocity components in the Cartesian x and y directions,

respectively. The matrix coefficients are computed from the mapping derived

from the exact solution and the contravariant velocity components are

U = U(b + v* and V = ml) + viJ; .
^x ^y X ^y

135

The physical boundary conditions for this problem represent subsonic
inflow at the entrance of the duct (at the lower left of Figure 7), supersonic
outflow at the exit, and the sides are treated as Impermeable boundaries
(walls). So that the Initial boundary value problem Is well-posed the
boundary conditions must be chosen carefully. See Kopriva, et al., [12] for
details of the procedure which follows. For the subsonic Inflow, we can
specify only two quantities and have chosen the total enthalpy and the angle
of the flow (so V = 0). The quantities P and U are computed from two
conditions: The first is a compatibility equation derived from the pressure
equation and the normal momentum equation. The second comes from
differentiating the enthalpy equation in time. From U and the condition
V = 0, the Cartesian velocities u, v can be computed. At the outflow, no
boundary conditions are needed. Finally, at the walls the normal velocity, U,
must vanish. The vector Q is computed by solving the tangential momentum
equation for V and a compatibility equation which combines the normal
momentum and pressure equations for P.

The system of equations (22) were discretlzed as described above, and
fourth-order Runge-Kutta was used for the time integration. For a single
domain, the Chebyshev spectral grid for the Ringleb problem with 16 streamwise
and 8 normal mesh intervals is shown in Figure 8. It is clear that the
spectral method strongly concentrates the grid points near the walls. The
largest gradients, however, occur in the streamwise direction near the sonic
line (as can be seen in Figs. 7 and 9) where the streamwise mesh distribution
is coarsest. These two factors contribute to the fact that the time
integration step is very small and that accuracy is degraded by the lack of
resolution where it is needed.

136

A multldomain grid distribution for which performance will be compared to
the single domain method is shown in Figure 10. Six domains now cover the
duct and the same number of mesh intervals as for the single domain case are
used. The divisions were chosen to demonstrate the kinds of situations which
the multidomain method should be able to handle. Three divisions with
6 + 5+5 mesh intervals are in the streamwise direction and two are in the
normal direction. With this choice, two points occur where the corners of
four domains come together. The first domain boundary in the streamwise
direction was chosen to appear in a subsonic region of the duct. The second
domain boundary in the streamwise direction was chosen to intersect the sonic
line. By dividing the normal direction into two domains, the effective mesh
spacing near the walls is doubled. Finally, note that by comparing Figure 10
to Figure 7 the sonic line also intersects the domain interface in the normal
direction.

To allow comparison. Figure 11 shows the Mach number contours for both the
single domain and the multidomain solutions. Note particularly that the sonic
line remains smooth through the domain interfaces. Table IV summarizes the
performance of the single domain spectral method compared with this particular
choice of grid. First, note that even with this distribution of domains, the
maximum error in the pressure for the multidomain computation has not been
degraded from the single grid one. In fact, the error is five percent better.

The real advantage that the splitting has had for this case, however, is
that the multidomain solution relaxes more quickly to steady state for a given
number of intervals and accuracy. Figure 12 compares the rate at which the
discrete L2 norm of the residual of the pressure decays for the single and
multidomain cases. The results are also summarized in Table IV. From the

137

trend of the graph, it should take over 2 1/2 times as many iterations for the
single grid residual to decay to that of the single grid residual. This is a

direct result of the fact that larger time steps can be used for the multi-

*
domain case. The choice of X also affects the convergence rate: larger

values up to the stability limit give faster convergence to steady state.

The advantage of a k-domain derivative computation requiring 1/k the
amount of work as a single domain computation does not show up in this
example. In fact, as Table IV indicates, the average time per Iteration (time
step) requires the same amount of time at 0.5 sec. on the Langley Cyber 855.
This is due to the fact that there is overhead in computing the interface
approximation. Doubling the number of points in each direction with the same
domain distribution decreases the time per iteration for the raultidomain
computation to 70% of the single domain cost. Though no attempt was made to
compute the interface conditions efficiently, the number of points inside each
domain will have to be large compared to the number of domains for the
efficiency gained by being able to use fewer points in computing derivatives
to become important.

The final advantage of a multidomain method which was listed in the
Introduction is that flexibility in the choice of grid point distribution is
now possible. A series of calculations were made with the duct being divided
into two domain intervals in each direction. As with Figure 10, the direction
across the duct was divided in half and the same number of mesh points was
used. In the streamwise direction, however, only one domain boundary was
inserted. This boundary was inserted in several places along the duct with
different numbers of points on either side.

138

Results of some of the computations are summarized in Table V. The
division is reported in terms of the fraction of the total variation of the
velocity potential along the length of the duct. The first entry in the list
places the division approximately near the bend of the duct where the
gradients of the solution are the highest. It is clear that with a proper
choice of grid it is possible to obtain better accuracy with the multidomain
distribution of a given number of grid points than with a single grid. For
the best case computed here, the error is about 2 1/2 times better for the
multidomain calculation.

The problem of how to properly distribute points and subdomains in general
is a major one and is beyond the scope of this paper. If they are poorly
placed the error can be worse than the single domain error (see Table V). For
now, it is not known how to obtain the optimal point and subdomaln
distribution. Rather, some knowledge of the behavior of the solution must be
used as a guide.

CONCLUSIONS

We have described a simple approximation which allows a multidomain
spectral solution of quasilinear hyperbolic equations. Numerical examples of
linear equation models and ideal gas flow show that the method gives
advantages in both accuracy and efficiency over using a single domain.
Dividing up a computational domain into several subdomains gives the
possibility of local refinement and allows some flexibility in the
distribution of mesh points. It is possible to obtain better accuracy by
doing so. Also, with multiple domains it is possible to take larger time

139

steps than with a single domain. This increases the efficiency for using time
relaxation to acheive steady state solutions.

The use of a multidomain technique is also appropriate if discontinuities
are fitted as boundaries. When shocks occur within a flow, subdomains would
be arranged so that each shock lies on a subdomain boundary. In smooth parts
of the solution, the technique described here would be used. Along shock
Interfaces, a shock fitting algorithm like that described in reference [1] can
be used (Kopriva and Hussaini, to be published).

The theoretical issues which remain are many. Some theory for the range

*
of values which X can take for the method to be stable must be found.

However, choosing X to be the average of the largest and smallest

eigenvalues of the coefficient matrix has always worked. Finally, like the

problems associated with the p- version of the finite-element method, the

choice of domain and point distribution for a given number of points is an

open issue.

ACKNOWLEDGEMENTS

The author would like to thank Dr. S. F. Davis and Professor
L. N. Trefethen for helpful comments and suggestions, and the Massachusetts
Institute of Technology for computer equipment used in the course of the
investigation.

140

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11. D. Gottlieb and S. Orszag, "Numerical Analysis of Spectral Methods:
Theory and Application," SIAM, Philadelphia, 1977.

12. D. A. Kopriva, T. A. Zang, M. D. Salas and M. Y. Hussaini,
"Pseudospectral solution of two-dimensional gas-dynamic problems" in
Proceedings of the 5th GAMM Conference on Numerical Methods in Fluid
Mechanics , (M. Pandolfi and R. Piva, Eds.), Vieweg, 1984.

13. R. Courant and K. 0. Friedrichs, Supersonic Flow and Shock Waves , New
York, Springer-Verlag, 1976.

142

TABLE I. L2 errors for the solutions to Eq. (20) with equal
number of points on each side of the interface.

N

8

16

32

Error in u

1.57 X 10
4.15 X 10
1.91 X 10'

-2
-6

Error in

1.49 X 10
4.86 X 10
1.91 X 10

-2
-6
-9

TABLE II. Lo errors for the solutions to Eq. (20) with more
points to the right of the interface.

\^\

8, 16
12, 24
16, 32

Error in u

1.22 X 10
2.45 X 10
3.93 X 10

-2
-4
-6

Error in v

1.05 X 10
2.33 X 10
3.93 X 10

-2
-4
-6

TABLE III. L2 errors for the solutions to Eq. (20) with
more points to the left of the interface.

Nl'Nr

Error in u

16, 8

9.80 X 10

24, 12

3.48 X 10'

32, 16

1.49 X 10

-4
-6

Error in v

1.04 X 10
2.88 X 10
2.30 X 10

-2
-4
-6

143

TABLE IV. Performance comparison for single and multidomain spectral

computations.

Grids: Single Domain (SD) 17 x 9 points

Multidomain (MD) (7 + 6 + 6) x (5 + 5) points
(separated by domain)

Maximum Error
SD
MD

1.85 X 10
1.74 X 10

-3
-3

Number of Steps to Reduce Residual Three Orders of Magnitude
SD > 1500

MD 780

Average Spectral Radius

SD 0.9964

MD 0.9942

Average Time per Iteration

SD 0.50 sec.

MD 0.50 sec.

TABLE V. Effect of streamwise mesh distribution
on Ringleb calculation.

Grid

Division

Maximum Error

8 + 8

0.45 + 0.55

7.8 X 10

8 + 8

0.50 + 0.50

9.3 X 10

16(SD)

1.9 X 10

10 + 6

0.34 + 0.66

1.2 X 10

-4

-3
-2

144

T

^_

^

(N

r

: = :

.

-(.

^

ITdTT

v

hV"?;-

:::

:==!=

=h=

— 1

FIG, 1. Diagram of the two-dimensional subdomain structure used to divide a
computational domain.

145

60

30

-30

-60

i

-90

-60

-30
RE

o

30

FIG. 2a. Effect on the eigenvalues of the two domain spatial approximation of

*

the first derivative by varying X in the boundary approximation:

*
X = 0.

146

60

' 1

30

- O

o

o °
o

—

o

\

O

o

o ^

o

o

•30

- o

•60

-90

-60

-30
RE

30

FIG. 2b, Effect on the eigenvalues of the two domain spatial approximation of

*
the first derivative by varying X in the boundary approximation:

X = 0.5.

147

60

30

-30

-60

-90

-60

-30
RE

o
o

o
-e-

o
o

30

FIG. 2c. Effect on the eigenvalues of the two domain spatial approximation of

it

the first derivative by varying X in the boundary approximation:

it

X =1.0 (purely upwind).

148

60

1 1

o

30

o

o
o

r\

o o

\

KJ

o o

o
o

o

30

o

60

1 1

-90

-60

-30
RE

30

FIG. 2d. Effect on the eigenvalues of the two domain spatial approximation of

the first derivative by varying X in the boundary approximation:

*
X = 1.1.

149

60

1 1

o

30

O

o o

oo

^

oo

)

o o

o

30

o

60

1 1

-90

-60

-30
RE

30

FIG, 2e. Effect on the eigenvalues of the two domain spatial approximation of

the first derivative by varying X in the boundary approximation:

*
X = 5.0.

150

ZD

FIG, 3a. Solution of the scalar pulse problem Eq. (19) computed on two
domains shown after the pulse has travelled from the left through
the interface at x = 0. Computations are for 22 points left and 11
points right- of the interface. The exact solution is the solid
line; computed solutions are the circles.

151

FIG. 3b. Solution of the scalar pulse problem Eq. (19) computed on two
domains shown after the pulse has travelled from the left through
the interface at x = 0. Computations are for 11 points left and 22
points right. The exact solution is the solid line; computed
solutions are the circles.

152

LU

CD
O

1 1 1 1

i 1 1

-1

—

—

-2

—•

—

-3

t

h— x:

-I'

y^^^^^^^ 0000000 ood

^*rl»

1

-5

Bs^^-B^B anaaaDDa gpna

—

-6

—

—

-7

—

—

-8

-9

-10

—

—

ktA-/^

1 1 1

-2.0 -1.5 -1.0 -.5

X

.5 1.0 1.5 2.0

X

FIG. 4a. Pointwise errors as X varies for the situation in Figure 3a.

153

bJ

CD
O

-10

FIG. 4b. Pointwise errors as X varies for the situation in Figure 3b.

154

-2

-1

X

FIG. 5a. Graph of the two solutions u (circles) and v (squares) of the
linear system Eq. (20) with 22 points on the left and 11 points on
the right. The exact solutions are represented by the solid line.

155

X

FIG. 5b. Graph of the two solutions u (circles) and v (squares) of the
linear system Eq. (20) with 11 and 22 points on the left and the
right. The exact solutions are represented by the solid line.

156

.2

.6

.8 1.0 t.2 1.4 1.6 1.8 2.0 2.2 2.4 2.6 2.8 3.0

FIG, 6. Plot of the computed pressure in a converging-diverging nozzle where
the interface is placed at the sonic point at x = /2. Twice as
many points are used on the right as on the left of the interface.

157

FIG, 7. Mach contours of the exact solution to the Ringleb problem which
models transonic flow in a two-dimensional duct.

158

FIG. 8. Single domain Chebyshev grid for the Ringleb problem.

159

2.0

o

en

-1.65 -1.10

X

FIG, 9. Mach number variation along the lower wall, center streamline and
upper wall for the Ringleb problem.

160

FIG. 10. Multidomain grid with six subdomains for the Ringleb problem.

161

FIG, 11a. Mach number contours for single domain solution.

162

-44-

FIG. Hb. Mach number contours for six domain solution.

163

-45-

cc

ID
CD

-1 -

CO
LU

LD
O

-2 -

-3 -

-4 -

-5

500 1000

ITERATION

1500

FIG. 12. Comparison of residual decay for single domain and multldomaln
solutions to the Ringleb problem.

164

ON SUBSTRUCTURING ALGORITHMS AND SOLUTION TECHNIQUES
FOR THE NUMERICAL APPROXIMATION OF PARTIAL DIFFERENTIAL EQUATIONS

M. D. Gunzburger
Carnegie-Mellon University

R. A. Nicolaides
Carnegie-Mellon University

ABSTRACT

Substructuring methods are in common use in structural mechanics problems
where typically the associated linear systems of algebraic equations are
positive definite. Here these methods are extended to problems which lead to
nonpositive definite, nonsymmetric matrices. The extension is based on an
algorithm which carries out the block Gauss elimination procedure without the
need for interchanges even when a pivot matrix is singular. Examples are
provided wherein the method is used in connection with finite element
solutions of the stationary Stokes equations and the Helmholtz equation, and
dual methods for second-order elliptic equations.

Support for the first author was provided by the Air Force Office of
Scientific Research under Grant No. AFOSR-83-0101 and by the Army Research
Office under Contract No. DAAG-29-83-4-0084. The second author was supported
under AFOSR Grant No. AFOSR-83-0231. Additional support was provided by the
National Aeronautics and Space Administration under NASA Contract No. NASl-
18107.

165

1. THE SUBSTRUCTORING ALGORITHM IN THE POSITIVE DEFINITE CASE

The use of substructuring techniques in the numerical solution of problems
governed by positive definite partial differential equations is in widespread
use. The most notable case is found in structural mechanics, especially in
connection with the equations of linear elasticity. For the sake of
simplicity, here we describe the technique for the Dirichlet problem for the
Poisson equation. Specifically, suppose we want to solve

-Au = f in n

u = on 3fi

(1)

where n is, say, an open bounded region in IE? with boundary 9n. We

subdivide the region fi into open subregions fi , i = l,««»,m, such that
m

n = U JT. and n Hfi . = for i * j. We denote by r. .,l<i<i<m
1=1 1 1 J ij - -

the interfaces between regions n and a , i.e., r.. =?r.Pin'.. Of course,
for particular choices of i and j in a given subdivision, r may be
empty. A sketch of a particular example with m = 5 is given in Figure 1.

Figure 1. A subdivision of a region into five subregions.

166

We also subdivide n Into a finite difference or finite element grid
which in practice is much finer than the above subdivision of fi into m
subregions. We choose the two subdivisions so that the Interfaces r. .
coincide with edges of the finite difference or finite element cells. The
discretization of (1) proceeds in the usual manner. The essence of the
substructuring algorithm is found in the particular choice for the ordering of
the unknowns and equations, i.e., columns and rows, in the linear system
resulting from the discretization of (1). Specifically, all unknowns and
equations associated with the interior of a substructure Q. are numbered
sequentially, one substructure at a time, and unknowns and equations
associated with the Interfaces F . . are grouped together and numbered last.
For example, in a typical finite difference discretization of (1), one
associates equations and unknowns with nodes in the grid. In this case, we
would group together all the unknowns in subregion Q^ together and number
them first, then proceed to f2„ , etc., and finally to Q . Then we would
number all the unknowns along the Interfaces T , . 1 <^ 1, j _< m. The
equations would be numbered in the same way. Likewise, in a finite element
discretization of (1), some unknowns (trial functions) and equations (test
functions) are associated with nodes or edges and these are

The subdivision and numbering method described here applies to difference
methods with stencils Involving only nearest neighbors. The method may be
extended in an obvious manner, e.g., by defining the interfaces to be more
than one grid point in thickness, to methods having stencils with a greater
degree of connectivity.

167

numbered in the same manner as in the finite difference case above. ^ In
addition there may be test and trial functions more naturally associated with
the finite elements themselves, and the equations and unknovms associated with
these functions are grouped together with the other ones associated with the
interior of the corresponding subregion fi .

The net result of the above numbering schemes is that the linear system
resulting from the discretization of (1) has the form

^2

A
m

C„ • • • C
I m

(2)

In (2), the matrices A . , i = l,»»«,m, in the finite element case, result in

the case of both the test and trial functions being associated with the

interior of the subregion n , i = l,««»,m, respectively, while the matrix

Aq results from test and trial functions associated with the interfaces r. . .

ij

1 <. i < j <_ m. The matrices Cj^, and B^^, represent trial, respectively test.

2
Again, the method described here applies to the case where the test and trial

functions vanish outside the elements which contain the associated node or

edge. However, by defining the interfaces to be one or more elements thick,

the method may be easily extended to other cases, e.g., cubic B-spline test

and trial functions.

168

functions associated with the interior of ^ and test, respectively trial,
functions associated with the interfaces. The vectors 11^,1= l,»»«,m,
respectively denote the unknowns associated with the interior of n, ,
i = l,»»«,m, while Uq denotes the unknowns associated with the interfaces.
All of these associations can also be made in the finite difference case.

It is well-known that the coefficient matrix of the linear system (2),

resulting from a discretization of (1), is symmetric and positive definite.

T T T

Indeed, A. = A , B, = C for i = l,«»»,m and A_ = A„. It is also easy to

see that the matrices k^, i = l,«««,m, are themselves positive definite. In

fact, these matrices are exactly the ones which would result from the

analogous discretization of the problems

Au = f in n

(3)
u = on 9a

for i = l,»»«,m, where 3n. denotes the boundary of n . . Note that this
boundary may consist of both interfaces and a portion of the boundary 3f2 of
fj, as is the case for n. , n» , fJ, , and ^c in Figure 1, or may consist wholly
of interfaces as is the case for fi, in that figure. Discretization of (3)
results in a linear system with a coefficient matrix A^, and thus A^ is
clearly symmetric and positive definite. We note that even in the case of the
Neumann problem, i.e., the boundary condition in (1) is replaced by 3u/8n =
on 9n, the matrices A^ in (2) would still be, at least in the finite
element case, symmetric and positive definite. This is so because the
problem (3) associated with the matrix A^ is now given by

169

Au = f in n

1^ = on an.nan W

9n 1

u = on 9^j^nr^, , j = l,««»,m,

where we have set Y .. - Y ... Since 8n.Or . is never empty, the matrix
A^ associated with (4) is symmetric and positive definite.

With the matrices A , i = l,«'»,m, being positive definite, one may
proceed to solve (2) by a block elimination procedure. Symbolically, we may
express the first m stages of this procedure by the relations

Uj^ = A^^(F^ - B^ Up), i=l,.-,m, (5)

which uniquely express U^ in terms of data and the interface unknowns Uq.
The last stage of the process requires the solution of the linear system

DU = G (6)

where

° = ^0 - J^ 'i ^i' ^ ""' ^ = 'o - J^ ^i ^i' "i-

(7)

^If on 8fi.03J^ something other than Dirichlet data is specified, then the
matrix A^ also contains rows and columns associated with test and trial
functions associated with nodes or edges on that portion of the boundary.

^Of course, the fact that A^, i = 0,...,m, are positive definite may be
deduced directly from the fact the coefficient matrix of (2) is positive
definite, i.e., the former is a necessary condition for the latter.

170

Of course. In (5) and (7) the inverses are not explicitly computed, but rather
appropriate linear systems are solved. The solvability of the system (6)
follows whenever the system (2) is solvable. In fact, if the system (2) is
positive definite, so is the matrix D [1], Once (6) is solved for Uq, (5)
yields U^, i = !,•••, m.

Although we have described the substructuring algorithm in the context of
the Poisson equation, the method can be applied in a similar manner to any
positive definite problem. As noted above, the method has encountered great
success in structural mechanics problems. However, in other fields where the
governing equations are not positive definite or symmetric one may still order
the equations and unknowns to produce linear systems such as (2), but these
may not always be solved by a standard block elimination procedure. In the
next two sections we describe a procedure to solve (2) even in the case of the
matrices A^ being singular and show how the method may be implemented
through an elimination procedure. In Section 4 we describe examples which
lead to singular matrices A^, Finally, in Section 5 we give some concluding
remarks.

Incidentally, in almost all situations the use of a properly implemented
substructuring algorithm will result in savings in computational costs when
compared to a banded elimination procedure. For example, consider a
discretization of Poisson's equation on a unit square. Suppose we have M
subregions in each direction so that ra = M and suppose that each subregion

is further subdivided- by introducing an n x n grid. Thus, there are a total

4 4
of Mm points in each direction. Banded elimination requires 0(M n )

operations, while the above substructuring algorithm can be implemented in, at

4 4 3
most, 0(Mn + M n ) operations. We note that this particular problem is not

171

particularly well-suited for substructuring methods. Also, the relative
advantage of substructuring is greater when one considers three-dimensional
problems or systems of partial differential equations.

We also note that substructuring ideas in connection with preconditioning
techniques have been discussed in [2].

2. THE SOLUTION ALGORITHM IN THE GENERAL CASE

We begin by describing a method for solving (2) in the case where the
matrices A^ are singular. The algorithm described here is a special case of
a more general algorithm which applies to arbitrary matrices with arbitrary
subdivisions into blocks, e.g., the matrix has no special structure and the
matrices A^^ may not only be singular, but may even be rectangular. The more
general algorithm is described in [3]. We will describe the algorithm as
applied to (2) and we will make use of pseudo-inverses in order to simplify
the initial presentation. However, we emphasize that the algorithm may be
implemented without the need for the explicit calculation of any pseudo-
inverses; such an implementation is discussed in the next section. This is
similar to the observation that the algorithm contained in (5)-(7) may be
implemented without explicitly computing any inverses, e.g., by solving linear
systems.

The system (2) is equivalent to

^i "i ■*■ ^i "o " ^i* ^ '^ 1, •••,!", (8)

m

l^ C^ U^ + Aq Uq = Fq. (9)

172

Now, Uj may be orthogonally decomposed in the form

where

U^ = Y^ + Z^, i = l,-«-,m, (10)

k^Z^= 0, i = !,•••, m, (11)

and X^ is orthogonal to all vectors satisfying (11). In particular,

Y^ Z^ = 0, i = l,«",m. (12)

Substitution of (lO)-(ll) into (8) yields that

A^ Y^ = F^ - B^ Uq, i = l,.--,m. (13)

Since Yj|^ is orthogonal to the null space of A^, (13) yields that

Y^ = A^(F^ - B^ Uq), i = l,.-,m, (14)

where A. denotes the pseudo-inverse of A^. This relation states that Y^^
is uniquely determined from the data and Uq. Note that (8) yields no
information concerning Z^ as is to be expected since A. Z^ = 0.

Substituting (10) and (U) into (9) yields that

m
DU„ = G - y C, Z, (15)

° i=l ^ ^
where

173

m , m

D = A - I C A+ B and G = F - f C A^" F . (16)

1=1 ^ ^ u ^^^ 1 1 1

We may also decompose Uq In the form

where

Uq = Yq + Zq (17)

DZq = (18)

and Yq Is orthogonal to all vectors satisfying (18). In particular,

yJ Zq = 0. (19)

Substitution of (17)-(18) into (15) yields that

m
DY = G - I C Z (20)

1=1

and, since Yq is orthogonal to the null space of D, (20) yields that

Yq = d"^(G - I C Z ). (21)

1=1

Again, it is not surprising that (15) yields no information concerning Zq.

Substitution of (17) and (21) into (14) then yields that

h = A^[F^ - B^ D'iG - I C. Z.)] - A^ B^ Z^ (22)

for i = 1, • • • ,m.

174

At this point we have shown that Y^, i = 0,«««,m, may be uniquely
expressed in terms of Z^^, i = 0,'«',m, by (21) and (22). It remains to show
how to find the latter. The first step is to multiply (13) by (l - A A^) .

Since A, A, A = A , we have that

(I - A^ A^)(F^ - B^ Uq) =0, i = l,-..,m,

or substituting (17) and (21),

(I - A^ A^)[F^ - h^Q- h ^^"'(G - I C Z

j = l -^ -^

m

)] = 0, i = l,...,m. (23)

Now suppose we are able to determine bases for the null spaces of Aj,
i = l,»««,m, and D. We collect each of these basis sets into matrices Nj,
i = 0,«»»,m, i.e., N^, i = 0,»»«,m, have linearly independent columns.

DNq =0 and A^ N^ = 0, i = l,...,m, (24)

and the columns of Nq, respectively N^, span the null space of D,
respectively A^, i = l,»»»,m. The number of columns in Nj is, of course,
the dimension of the corresponding null spaces. Now, we may write that

^1 " ^i ^1' ^ = 0,...,m, (25)

for some vectors A . Substituting (25) into (23) then yields that

m

I R^ A = H^, i = l,...,m, (26)

j = l -^ -^

175

where

ho = (I - A^ a\)b^ Nq. H^ = [I - A^ A^)[F^ - B^ d"" G]
and (27)

R. , = fl - A. A,)b, d"*" C. N., j = !,•••, m.

ij

Now letting

% ^1

^20 ^21

R =1 • • . I, H = I . I and A =1 . I , (28)

mO ml

(26) may be expressed In the form

RA = H. (29)

In general, R is a rectangular matrix. The number of rows in R is equal to
the sum of the number of rows of the matrices Aj, i = l,«««,m, and the number
of columns of R is equal to the sum of the dimensions of the null spaces
of Aj, i = l,»»»,m; and D. It can be shown [3] that the system (29) is a
consistent system, and we may find its solution, for example, by forming

(r'^R)A = r'^H. (30)

Suppose we can solve (30) for A. Then (28) yields A. , i = 0,««»,m, (25)
then yields Zj^, i = 0,»'»,m, (21) and (22) yields Y^, i = 0,»'«,m, and
finally (10) and (17) yield the solution Uj^, i = 0,..',m, of (2).

176

The algorithm described here is related in the following manner to the
block elimination algorithm in Section 1. Suppose that the matrix of (2) and
all the A^'s and D are nonsingular. Then, the algorithm of this section
reduces to the standard block Gauss elimination procedure. Indeed, in this
case, A^ = A^ , D = D and Z^ = so that U^^ = Y^ and the latter are
determined uniquely by (14) and (21). Note the correspondence, in this case,
between (14)-(15) and (5)-(6).

In the more general case, i.e., some or all of the A.'s and D being
singular, it can be shown [3] that the rank deficiency of (30) is exactly that
of the original coefficient matrix in (2). Therefore, if the latter is
nonsingular, then so is R^R and then A in (30) is uniquely determined.
Since the Z^^'s and Y^'s are uniquely determined from A, the algorithm
produces the unique solution of (2). If the matrix of (2) is singular, so
is R R and (30) does not have a unique solution. However, (30) may be
solved anyway, either in terms of arbitrary parameters or by adding
constraints. The number of parameters or constraints is equal to the
dimension of the null space of 'sJr which in turn is the same as the
dimension of the null space of the coefficient matrix in (2). In any case,
once a particular A is determined, then Z^ and Yj_ are also determined.

In particular applications to the solution of partial differential

equations, the dimension of the system (30) is small compared to that of the

T
system (2). Indeed, typically dim(R R) = 0(m) , the number of subregions.

For example, the dimension of the null spaces of the matrices A^ and D may

T
be one or zero, in which case dim(R R) < m + 1.

177

3. AN ELIMINATION IMPLEMENTATION

We begin by restating the algorithm of the previous section. Given the
matrices Aq,'«',A , B. .•••,B , C ,«««,C^ and the vectors Fq,«««,F^, we find
vectors U„,««»,U satisfying (2) by the following procedure.

1. Compute A, F. and A, B. for i = l,»»«,m.

2. Compute N , , i = !,•••, m, whose columns constitute a basis for the null
space of A , i = l,«««,m, respectively.

3. Compute C (A^ B ), C (A^ F ) and C. N, for 1 = l,«««,m.

m . m

4. Compute D = A - I C (a;J B ) and G = F - I C (a;|: F ).

u ^^^ 1 1 1 U ^^j X 1 1

5. Compute D"*" G.

6. Compute Nq whose columns constitute a basis for the null space of D.

7. Compute D C, N. for i = l,«««,m.

8. Compute the matrices

^iO ° ®i ^0 " \ ^4 \^^0 ^°'' i = l.***''">

hi = B^tD"*" C. H.) - AjA^ B^)(d'' C. Nj) for i,j = 1,...^

m

and the vectors
H^ = F^ - B^(d"^ G) - A^(A^ f) + A^(A^ B^)(d"^ G) for i = l,...,m.

9. Assemble the results of step 6 into the matrix R and vector H

T T

according to (28) and then compute R R and R H.

T T

10. Solve the linear system R RA = R H for A and then compute A.,

1 = 0,*««,m, according to the partition of (28).

11. Compute Z. = N, A. for i = 0,«**,m.

178

+ " + ™

12. Compute Y = (D G) - I C, N, A, = (D G) - J C, Z, .
u j_i ill '' i i

m m

I C N A = (d"^ G) - I

i=l ^ 1=1

13. Compute Uq = Yq + Zq.

14. Compute Y^ = (A^ F^) - (a^ B^)Uq for 1 = l,...,m.

15. Compute ^^ = ^^ + Z. for 1 = l,...,m.

Other than steps 1, 2, 5, 6, and 10, the above algorithm requrles only matrix
and matrix-vector multiplications. In this section we show how to carry out
the other operations required by the algorithm through an elimination
procedure. In particular, we will not need to explicitly calculate any
pseudo-inverses of matrices.

We first describe how to carry out steps 1 and 2. Consider the linear
system.

A^ Q = S= (B^,F^,0) (31)

where the right-hand side matrix S consists of the matrix B^^, the vector
F^, and some additional columns of zeroes. The number of these additional
columns should be greater or equal to the dimension of the null space of
Aj^. This dimension will actually be determined during the elimination
procedure. 5 We now proceed to solve (31) by Gauss elimination with partial
pivoting. If the matrix A^ is singular, then one or more times during the
elimination procedure we will not be able to locate a nonzero pivot element.
In fact, the number of times this occurs is exactly the dimension of the null
space of A^. However, at such an occurrence, the corresponding column Is

See Section 5 concerning the effects that roundoff errors may have on the
determination of this dimension.

179

already in the eliminated form so that we may skip over to the next column and

continue the elimination process. At the end of the process, (31) has been
reduced to the form

A . Q = J = [\,F^,0) (32)

where

A. is upper triangular and in row echelon form. When Aj^ is

singular. A, will have zeros at the pivot location for exactly those columns
for which no nonvanishing pivot element was found.

We now proceed to backsolve (32). No difficulty is encountered until a
row is reached for which the pivot entry of A. is zero. For the columns
of Q corresponding to B^ and F^, we may arbitrarily set (to something
other than zero) the entry in the row corresponding to the zero pivot of A,.
Then the backsolve procedure may continue until we reach another zero pivot
entry, at which time we again arbitrarily specify an entry in the columns of
Q corresponding to the columns B^ and F^ of S, While all this is going
on we are also solving (32) for the columns corresponding to the zero columns
of S. For these columns, whenever a zero pivot entry is encountered in A.,
one of the elements in the corresponding row is set to one while the rest are
set to zero. Each time a zero pivot entry is encountered, a different column
is chosen for which one sets the arbitrary element to one. At the end of this
backsolve procedure, (32) yields that

Q = (L,K,N^).

Here the columns of Nj form a basis for the null space of Aj_ and L and

A

K are particular solutions of the systems.

180

A^ L = B^ and A^ K = F . (33)

The final step Is to orthogonallze the columns of L and K with respect
to the columns of N^ to yield

Q = (L,K,N^).

Since A^N^ = 0, L and K are still solutions of (33). Moreover, the
columns of L and K are orthogonal to the null space of A^ and,
therefore, are minimum norm solutions. By the uniqueness of the minimum norm
solution, we have that

L = A^ B^ and K = aI" F .

•Thus the above elimination procedure has accomplished the tasks of steps 1 and
2 of the algorithm.

The tasks of steps 5, 6, and 7 can be accomplished in an analogous
manner. Also, if the matrix r'^R is nonsingular, then it may be easily
solved by an ordinary Gauss elimination procedure. If it is singular then a
solution in terms of arbitrary parameters may be determined in a manner
similar to the above procedure for the system (31). We note that any sparsity
or structure inherent in the matrices A^ may be exploited in the above
procedure. However, in general, the matrix D will be dense. We will return
to this point in the concluding section.

181

4. EXAMPLES

The Stationary Stokes Equation

Consider the stationary Stokes equations for the slow flow of a viscous
fluid in a bounded region in Wr. These are given by

A_u - grad P = _£ in H

div u = in 9. (34)

u = on 9n.

Here ii denotes the velocity, p the pressure, f_ the given body force and
the viscosity coefficient has been absorbed into p and f^. Clearly, the
pressure cannot be determined uniquely since we may add an arbitrary constant
to the pressure and still satisfy (34).

A finite element approximation of the solution (u^,p) of (34) may be

Vi 1^

defined as follows. Given finite-dimensional spaces V and S for the
discrete velocity and pressure fields, we seek u eV and p eS such that

/(grad u : grad v - p div ^ )dn = -/ _f»v dQ for all _v €V

q*^ div u^ dJ2 = for all q'^eS^.

(35)

Here we assume that the elements of V satisfy the boundary condition In
(34). By choosing bases for the spaces V" and S", (35) can be expressed as
a linear algebraic system for the coefficients in the basis function
expansions of u and p .

182

Now it is well-knovm that arbitrary choices of spaces V and S may
not yield stable or accurate solutions. However, there are now known many
element pairs for which (35) yields optimally accurate solutions [4], [5],
[6]. One such pair is described as follows. Suppose S, denotes a
triangulation of the region n. We denote by V, a finer triangulation
derived from S. by subdividing each triangle in S. into four congruent
triangles by joining thp midsides. See Figure 2. We define S^ to consist
of piecewise constant functions over the triangulation S,

Figure 2. A triangle in S^ and the corresponding triangles in V, .

and V to consist of piecewise linear functions over the triangulation V,

h

which are continuous over Q and vanish on dQ. This combination is known to

be stable and be optimally accurate [6]. The basis functions for V^ are

easily associated with the vertices of the triangulation V^ while the basis

h

functions for S" are associated with the triangles in the triangulation S. .

See below for the necessary restriction on the pressure which yields this
result.

183

Now let us consider a substructuring technique for the solution of (35).
We assume that the interfaces T. . between subregions are made up of edges of
the triangulation S, so that these interfaces do not cut across pressure
triangles. One may easily arrange a numbering scheme for the unknowns and
equations which yields a linear system of the form (2). For example, Uj
consists of all velocity unknowns associated with vertices of V, located in
the interior of the subregion Q. and all pressure unknowns associated with
the triangles of S, which are also in U . Note that Uq contains only
velocity unknowns, namely those associated with vertices V, which lie on the
interfaces F , . but not on 9n.

We have not constrained the pressure space and therefore the system (2)

corresponding to this discretization of (34) is singular. In fact, its rank

defficiency is one, and the null vector corresponds to the pressure function

which is constant over fi. On the other hand, the velocity approximation is

uniquely determined by (2) [6]. Furthermore, it is easy to see that the

submatrices A, .•••,A are singular. In fact, these matrices are exactly
1 m

those which arise from the analogous discretization of the problem.

Au^ - grad P = f_ in fi.

dlv u^ = f in fi .

u = on 8n .

Thus each of the matrices k^ has a single local pressure null vector, i.e.,
the dimension of N^ is one and N^ corresponds to the pressure function
which is constant over fi . . On the other hand, since the velocity field can

184

be uniquely determined from (2) and since Uq consists of only velocity
unknowns, the matrix D in the linear system (15) is nonsingular, i.e.,
Nq = 0. Thus, in this case, the system (30) has dimension m and has a one-
dimensional null space, the latter following from the fact that the system (2)
Itself has a one-dimensional null space.

If we choose the pressure space S to consist of piecewise linear
functions over the triangulation S, which are continuous over Q, while
retaining the same velocity space, the situation changes drastically. For
example, now the basis functions for S are more easily associated with the
vertices of S, . Now Uj contains pressure unknowns corresponding to

vertices in S, which are in the interior of Q. or lie on 8J2.r^3fJ. More
n 11

important, Uq now contains pressure unknowns associated with vertices of S,
which lie on T . . but not on 9n. In this case the matrices Aj are
nonsingular and the matrix D is singular with a one-dimensional null space.

The Helmholtz Equation

Now consider the problem

Au + Xu = f in n

(36)
u = on 90

where X is not near an eigenvalue of the operator -A . Standard finite
element or finite difference discretizations of (36) yield linear algebraic
systems with coefficient matrices which are symmetric and indefinite, but
which certainly may, by using a partial pivoting strategy, be stably
inverted. Now consider the following specific situation. Let ^ be the

185

square (0,ir) x (0,ir) and let X = 13/A. Since the eigenvalues of -A for
this region are given by (n^ + m ), m,n = 1,2,»««, we see that X = 13/4 is
not an eigenvalue and therefore the problem (36) leads to nonsingular
coefficient matrices. Now, suppose we consider solving (36) by using the
substructuring algorithm with the two subregions n = (0,2Tr/3) x (0,tt) and
fj- = (2ir/3,Tr) Q (0,it). Then the matrices A^ in (2) correspond to the
coefficient matrix for the analogous discretization of the problem

Au + Xu = f in n

u = on 9fl . .

(37)

2 2
But the eigenvalues of -A for the region n. are given by (n + 9m /A),

m,n = 1,2,3,«««, so that X = (13/4) is an eigenvalue of -A for the region

Q, and therefore the matrix A^^ is singular even though the system (2) is

not.

Admittedly, this example is somewhat pathological in the sense that for

random choices of regions, subregions, and parameters X, the probability is

zero that the matrices A^ in (2) will b^ singular. However, for particular

choices of X, Q. and Q , , one or more of the matrices k^ may be singular;

after ^11, the above example is not really all that far-fetched. Of course,

if any of the A^'s are singluar, the situation may be remedied by choosing a

different subdivision of the region ^; this in turn implies a complete

reassembly of the coefficient matrix in (2). On the other hand, the algorithm

of Sections 2 and 3 may be used whether or not any of the matrices A^^ are

singluar.

186

There is a small but nonvanishing probability that for some of the
problems (37) X, although not an eigenvalue of -A for the region n. , is
close to such an eigenvalue. If X is close enough to such an eigenvalue,
the matrix A^, in finite precision arithmetic, may be mistakenly determined
to be singular by the algorithm of Section 3. However, this will be the case
only when the difference between X and an eigenvalue is much smaller than
the discretization error, i.e., of the order of the unit roundoff error of the
machine, and no serious effect on the accuracy of the solution should result.

Dual Methods for Second-Order Elliptic Equations

For a third example, we consider dual methods for second-order elliptic
partial differential equations. An example of these are methods based on the
complementary energy principle in linear elasticity. For simplicity, we here
consider the problem

u = V(|) in n

div u^ = f in fi

(38)
u'n_ =0 on r
and

(j) = g on r2

where again ^I'^^o ~ ^^ denotes the boundary of the bounded region fiClr
and n denotes the unit outer normal to 9n. A finite element approximation
of (38) may be obtained by choosing finite-dimensional spaces V and S
and then seeking u eV and (}> €S such that

187

J ^^ dlv u^ da = / f/ V.l,^€S^.

We assume that the elements of V satisfy the boundary condition on r. In
(38). The boundary condition on cj) is natural in this formulation, which is
one of its advantages.

In [7], the following choice of V and S" was shown to yield stable
and optimally accurate approximations, at least for polygonal domains. First,
we subdivide Q. into quadrilaterals, and then subdivide each quadrilateral
into four triangles by drawing the diagonals. For V we take all continuous
piecewise linear vector fields with respect to the resulting triangulation and
then define S" = div V". The resulting space S^ can be shown to be a
subspace of all piecewise constants over the triangulation. See [7] for
details.

In the implementation of the substructuring algorithm, we assume that the
interfaces r . . coincide with some of the edges of the quadrilaterals which
initially defined our finite element triangulation of fi, i.e., the interfaces
do not cut through any of these quadrilaterals. The test and trial functions
from V" are associated with nodes while those from S" are associated with
the interior of the quadrilaterals. The matrices Aj|^ in (2) now correspond
to the discretization of the problem

tj = V(j) and div u^ = f in Q..

(39)

u«n = on r nan,, i) = g on r-Hsn

188

and

u = on r. .nan..

- ij i

Because of the last boundary condition, the problem (39) is over constrained
insofar as the variable ii us concerned. Nevertheless, if r»08n. = 0,
i.e., a given subregion does not have part of its boundary coincide with that
part of 8n on which data for (j) are given, then the problem (39) can only
determine (j> to an additive constant. This, for example, would be the case
for subregion S^„ in Figure 1, i.e., an interior subregion. For such
situations, i.e., r„08n. = 0, the matrix A^^ in (2) will again be singular,
with a one-dimensional null space. Since (38) always uniquely determines ii,
the matrix D of (16) will be nonsingular. The rank deficiency of the system
(30) will be one or zero, depending on whether or not r„ has vanishing
measure, i.e., whether or not the problem (38) uniquely determines (j).

5. CONCLUDING REMARKS

Determination fo Zero Pivot Elements

A crucial step in the elimination algorithm presented in Section 3 is the
determination of when all the elements in a column to be eliminated are
already zero. This is necessary for the determination of the null spaces of
the matrices A^ and D. In practice one would declare an element to vanish
whenever its magnitude is less than some prescribed tolerance which should be
proportional to the unit roundoff error of the machine. This naturally leaves
open the possiblity of a very small but nonzero element being mistaken for a
vanishing element. This situation can be avoided, at least when one is

189

solving partial differential equations, by first using high enough precision
arithmetic, e.g., 60 or 64 bit floating point arithmetic, and by making sure
that the algorithms used are stable. The former Is easily arranged, while the
latter points out the Importance of rigorous mathematics. Indeed, If an
algorithm Is stable, as are the ones discussed In Section 4, and the machine
precision Is high enough, one should not encounter nonzero elements which are
comparable In magnitude to the unit roundoff error unless the matrix In hand
Is singular or very nearly singular.

An alternative to the use of elimination type procedures Is, of course, to
employ methods based on orthogonal transformations. At the price of greater
computational expense, such methods are less susceptible to 111 effects due to
roundoff error.

Parallelism

One of the attractions of substructurlng algorithms Is the obvious
Inherent parallelism both In the assembly and solution stages. The sets of
matrices and vectors (A^,B ,C^,F ), 1 = l,««»,m, can each be assembled
independently. Furthermore, at least in the finite element case, we may write
the matrix Aq and the vector Fq in the form

m m

^ = J^ ^01' ^0 = J^ ^01 (^o>

where the matrix Aq^ and the vector Fq^ represent the contribution to the
matrix Aq and vector Fq coming from region fj . Each of the sets (A„, ,
Fq^), 1 = !,•••, m, may be assembled in parallel. Thus, in the assembly stage,
the sets (A^,B^,C^,F^,Aq^,Fq^) , 1 = !,•••, m, may be assembled in parallel.

190

For example, each of the above sets may be assembled on separate processors,
with no need for interprocessor communications. At the end of the assembly
process, the concatenations of (40) must be performed. This step is not
parallelizable, but represents a minor portion of the assembly process.

There is also a large degree of parallelism in the solution algorithm
described at the beginning of Section 3. Steps 1, 2, and 3 are completely
parallelizable, again with no interprocessor communications necessary.
Furthermore, if the appropriate information can be transferred to the
processors, steps 7, 8, 11, 14, and 15 and a portion of step 12 can also be
computed in parallel. The only relatively major steps which are not
parallelizable are steps 5 and 6.

The issue of parallelism in connection with substructuring algorithms has
been studied in [8] in the context of a specific three-dimensional positive
definite problem. That paper contains a discussion of operation counts which,
for the most part, is also relevant in the present context.

Three-Dimensional Problems

As pointed out above, the major nonparallel steps in the computation are
embodied in steps 5 and 6 in the algorithm of Section 3. Even on a serial
machine these steps may be costly since, in general, they involve dense
matrices. In two-dimensional problems, by keeping the number of subregions
relatively small compared to the total number of elements in the
triangulation, the size of these dense calculations can be kept small, i.e.,
the size of D can be of the order of the square root of the size of the
Aj's. The latter usually are sparse, e.g., banded. A similar arrangement in
three-dimensional problems would, in general, lead to a matrix D whose size

191

is of the order of the two-thirds power of the size of the A^'s, which may be
unacceptably large. Furthermore, in steps 1 and 2 of the algorithm, the
number of right-hand sides would be approximately equal to the number of
columns of D and the size of the A^^'s may be too large, when relatively
few subregions are used. Therefore, for three-dimensional problems one must
be especially careful to implement the algorithm in an efficient manner as
possible.

These potential difficulties can be mitigated in a variety of ways. For
example, many of the right-hand sides in the computations of step 1 of the
algorithm are zero because any column of B^ which corresponds to an
interface unknown which is not associated with 8a. would vanish. The
corresponding row of C^ is also zero. Thus, one can avoid computations
involving linear systems with zero right-hand sides and multiplications by
zero vectors. The savings possible, in storage and computing time, by
accounting for these features are relatively higher for three-dimensional
problems.

Although, in general, the number of interface variables may be large for
three-dimensional problems, in practice it is often the case that specific
features of the domain n lead to a small number of such unknowns. For
instance, in a wing-fuselage configuration, it is natural to consider the wing
and fuselage to be different subregions and the interface between these two
substructures is relatively small in extent. Indeed, it was exactly in this
type of application that the terminology "substructuring" arose.

Finally we consider the most serious problem, namely that of the size of
the matrix D. However, even here a judicious implementation can effect great
savings. As a simple illustration consider the subregion structure of Figure
3 where we have now labeled the interface boundaries by r . , i = 1 , • • • ,m - 1.

192

Figure 3. An example subdivision of the region Jl.

It is natural to order the interface unknowns Uq one interface at a time,
e.g., first those on T., then those on r^, etc. It is not hard to see that
the matrix D for this example is block tridiagonal, i.e., the unknowns
corresponding to the interface V. are connected only to the unknowns on the
interfaces r ._. , r., and T . ' . By taking advantage of features such as
this, the cost of step 5 and 6 of the algorithm can be greatly reduced,
especially in three-dimensional settings. We note that these ideas are
similar to those connected with one-way direction algorithms for positive
definite problems [9].

193

REFERENCES

[1] F. Gantmacher, The Theory of Matrices , Chelsea, New York, 1960.

[2] G. Golub and D. Mayers, "Use of preconditioning over Irregular regions,"
in Computer Methods In Applied Science and Engineering , VI, (R. Glowlnskl
and J. Lions, Eds.), 1983, pp. 3-14.

[3] M. Gunzburger and R. Nlcolaldes, "Elimination with nonlnvertlble pivots,"
Linear Algebra Appl ., 64, 1985, pp. 183-189.

[4] V. Glrault and P. -A. Ravlart, Finite Element Approximation of the Navler-
Stokes Equations , Springer, Berlin, 1979.

[5] J. Boland and R. Nlcolaldes, "Stability of finite elements under
divergence constraints," SIAM J. Numer. Anal ., 20, 1983, pp. 722-731.

[6] J. Boland and R. Nlcolaldes, "Stable and semlstable low order finite
elements for viscous flows," SIAM J. Numer. Anal ., 22, 1985, pp. 474-492.

[7] G. Fix, M. Gunzburger, and R. Nlcolaldes, "On mixed finite element
methods for first-order elliptic systems," Numer. Math ., 37, 1981, pp.
29-48.

194

[8] L. Adams and R. Voigt, "A methodology for exploiting parallelism in the
finite element process," in Proceedings of the NATO Workshop on High
Speed Computations , (J. Kowolik, Ed.), Springer-Verlag, Berlin, 1984, pp.
373-392.

[9] A. George and J. Liu, Computer Solution of Large Sparse Positive Definite
Systems , Prentice Hall, Englewood Cliffs, New Jersey, 1981.

195

Multiple Laminar Flows Through Curved Pipes*

Zhong-hua Yang''' and H.B. Keller
Applied Mathematics, Caltech, Pasadena, CA 91125

Abstract

The Dean problem of steady viscous flow through a coiled circular pipe is
studied numerically for a large range of Dean number and for several coiling ratios.
We find that the solution family, as parameterized by Dean number, has numerous
folds or limit points. Four folds and hence five branches of solutions are found.
'We speculate that infinitely many solutions can exist in this family for some fixed
value(s) of D . More resolution and higher accuracy would be required to justify
our conjecture and to find the rule of formation of new solution branches.

*This work was supported by the "U.S. Department of Energy Office of Basic Energy
Sciences (contract DE-AS03-76SF 00767), and by the Army Research Office (con-
tract DAAG-29-81-K-0107). The calculations were done on the Caltech Applied
Math IBM-4341 supplied and supported by the IBM Corporation.

fPermanent address: Shanghai University of Science and Technology, Jiading,
Shanghai, China.

196

1. Introduction

Following the early work of Dean (1927, 1928) there have been several numerical
studies of the steady, laminar, viscous floA' of an incompressible fluid through a
slightly curved pipe of circular cross section. In particular, Dennis (1980) with
Collins (1975) and with Ng (1982) have computed such flows when the coiling ratio
a/L is small. Here a is the pipe radius and L is the radius of curvature of the
axis of the pipe. Also Van Dyke has applied the Stokes series and Dombes-Sykes
technique (1978) to this problem. In all of this work the crucial parameter is the
Dean number, D , defined as

D^Ga'i^f'/^tu (1.1)

where G is the constant pressure gradient driving the flow, /z is the viscosity and
u is the coefficient of kinematic viscosity. For small D and a/L « 1 all of the
results agree.

In particular for a straight pipe, a/L = 0, the flow is the classical Poiseuille
flow. However a slight curvature of the pipe axis iiiduces a centrifugal force on the
fluid which then forms a secondary flow, sending fluid outward along the symmetry
axis and returning along the upper and lower curved surfaces. Thus a pair of
symmetric vortices is superposed on the Poiseuille flow. These qualitative features
are observed in all of the previously cited references for D small and a/L << 1.
What happens as D and a/L increase? Few of the previous studies consider
a/L = 0(1). Further, Van Dyke's expansions disagree with the flnite difference
calculations for larger values of D. And in Dennis iz Ng (1982) dual solutions are
found for the range 957.5 < D < 5000 ; that is a four vortex solution is computed
in addition to the standard two vortex flow described above.

In this paper we attempt to clarify the situation by determining the structure
of the families of solutions that exist as D varies. In addition we show how this

197

structure changes as a/L increases (to 0.3). For this purpose we must retain the full
Navier-Stokes equations and do not make the ajL « 1 simplifications. However
no dramatic effects are found as a/L increases. Regarding the structure with
respect to D we are not completely successful. Our results suggest, in analogy with
the von Karman swirling flows (Lentini & Keller 1980), that there may be infinitely
many steady flows for some value (or interval) of D. However, we have found only
five branches of such flows and believe that more numerical accuracy is required to
completely settle the question. Indeed our first, cruder calculations revealed only
three branches of solutions. Unfortunately the variation in flow patterns from one
branch to the next are not as regular as those in the von Karman swirling flows,
so that we cannot have the same confidence in our current conjecture. Also, we do
not see analytical regularities in the five flows we have detected.

After our study was completed we learned of related calculations in curved
tubes by Winters and Brindley (1984) and by Winters (1984). However that work
is mainly concerned with tubes of rectangular cross section, with a brief mention of
the circular case in Winters and Brindley (1984). Bifurcations are obtained for the
rectangular case but they do not examine the results we study here.

In section 2 we formulate the problem retaining the exact equations (valid to all
orders in c = a/L ). Expansions in Fourier series are introduced in section 3 to get a
system of nonlinear two-point boundary value problems for the Fourier coefficients.
Numerical methods are introduced in section 4. These employ centered differences
and Newton's method with continuation or path following techniques introduced
by H.B. Keller (1977). The results are presented and discussed in section 5.

2. General Formulation

We employ the notation used in Collins & Dennis (1975) and Dennis & Ng
(1982) as indicated in Figure 1. The circular cross section of the tube in the [x, y)-
plane has radius a with center at L on the x-axis. The tube is coiled about a

198

circle of radius L in the (i, z)-plane. With no pitch in the coil the tube thus forms
a torus. Our equations are exact for this case. Dimensionless velocity components
of the fluid are (u,v,io) at a point P with dimensionless polar coordinates (r, a).
Here u is the radial and v is the angular component of velocity in the pipe cross
section, w is the axial velocity normal to the cross section and r = r' ja where r' is
the dimensional radius.

We seek flows independent of 6 , the angular deviation from the (z, y)-plane.
A stream function ^(r, ex) is introduced in terms of which the transverse velocity
components are given by:

uvr^a) = — ; r -T—

r(l + e r cos aj oa

-1 d<i>

v{r,a) =

(l + e r cos a) dr

(2.1)

Here c = a/L is the "coiling ratio" and the continuity equation is thus satisfied.
Using these velocity components in the Navier-Stokes equations we introduce the
modified Laplacian

V2 =

1 + e r cos a

' d , r d ^ d , e sin oc d ^

■ dr^l + e r cos a dr^ da ^\-i~'=-- '•<-><= '^ ^^^

+ e r cos a da.

(2.2)

and the vorticity

n = -v2,^,

(2.3)

to get for the u;-momentum equation

V^u; +

1 ,d4> dw d<f> dwy _

r(l + €. r cos a) dr da da dr '

and on elimination of the pressure from the other momentum equations:

(2.4)

199

r(l + er cos a) \'dr da da dr''

2e n ... d(j) cos a d4)s

+ t: r^ sin a -r— 4- -r— )

(1 + £ r cos o:j2 ^ or r da'

w . . dw cos q: SiWx

~ 77"; V2 ('5'"" ■^~ + T-j • (2.5)

(1 + e r cosa)^ ^ 5r r 5a' ^ '

The equations used in Dennis (1980) are obtained by setting e = in (2.l)-(2.5)
(i.e. they use the small coiling ratio approximation but we do not).
Boundary conditions on the wall of the tube, r = 1 , yield:

d^
da

w{l,a) = cf>{l,a] = — [l,a)=0 , < a < tt . (2.6)

We study here only flows symmetric about the x-axis for which:

w[r, a) = w[r, -a) , 4>[r, a) = -(f>[r, -a) , n(r, a) = -n{r, -a] . (2.7)

Thus on the symmetry axis we have:

dw , . dw ,
^(r,0) = -(r,.) = 0,

<^(r,0) = (;i(r,7r) = 0,

n(r,0) = n(r,7r)=0. (2.8)

3. Fourier Series Expansions

To solve the boundary value problem posed in (2.2)-(2.8) we seek Fourier ex-
pansions of the stream function, axial velocity and vorticity in the forms:

200

a) 4>{r,a) = ^./fc(r)sin ka ;

CO

b) w[r,a) = 2_[ Wk{r)cos ka ;

;c=o

CO

c) n{r,a) = ^ e;:(r)sm fca .

(3.1)

;c=i

With these forms the symmetry conditions (2.7) and the implied boundary condi-
tions (2.8) are satisfied.

Using the expansions (3.1) in the difTerential equations (2.3)- (2.5) and applying
the orthogonality properties and other identities for the trigonometric functions
yields an infinite system of coupled nonlinear, second order ordinary diff'erential
equations for the coefficient functions {/^(r), Wk{r), 5fc(r)} . Specifically we get
from (2.3), with the notation /o(r) = go{r) = :

er

rf2

d? (/: + l)(/: + 2) l

er

+ 7

dr^

= -y 9k-i{r) - gk{r) - — gk-^\[r) , k>l

From (2.4) we get, with w^i[r) = 0:

(3.2)

£r rd^ (fc-l)(fc-2)
2 idr^ r2

Wk~i{r) + [

d^ l_d_
dr- r dr

r2j

iyA;(r)

er

(fc + l)(/c + 2)

Ldr2

i«fc+l(r)

= -R;:(r)-6;c,ierr)-(5;,,oP, /c >

(3.3)

201

From (2.5) we get, with g-i{r) = 0:

.£r.2 rd^ {k-2){k-Sh (.,^n.d^ Id (/c-l)(2/:-3) i

er

+—

r ]^d?_ 1 _rf _ fc^]

rf2 1 d (A; + l){2A; + 3)'

eVrd^ fc^i

dr2

r^J

5/:(r)

-^ \gk+,[r) + (-) [^ ^^ i g,+2(r)

2 L rfr2 ' r dr

= 7 Sk-i[r) + 5;,(r) + ^ 5;:+i(r) + P;,(r) + 1 Q^r) , A; > 1 . (3.4)

We have used the Kronecker symbol 6i,y and introduced the quantities R},, S^,
Pk and Qk as:

a) R,[r)=^l±^Al g {[|n-/:|/,_,|(r) + (n + fc)/„+,(r)]<(r)

n=0 ^ •'

+ ^[fUk+ sign(n-/:)/|;_j^,(r)]u;^(r)j
^) ^'cW = 5; E [l^-^l /|n-;i|(0 - (" + fc)/n+.(r)lff;(r)

n=l *> ■'

- n[/;^;, - sign (n - k)f[^_j^^[r)^g^[r] I

c) -P/^C^) = 4^1 [<('•) - 7«^n(r)] [(1 + <5;,,„+i)u;|,+i_,|(r) - «;„+a+,.(r)

n=0 ^

- [<{r) + ^ w;n(r)] [(1 + 6n-i,k)vj\n-i.k\{r) - u;„_i+;,(r)] |

d) '?/:(r) = X;|[/;(r)-^/,(r)] L+i+;,(r) - sign (n + 1 - %|„+,_^,(r)l

n=l '^ J

- [/nW + ^ /n(r)] [pn-i+jt(r) - sign (n - 1 - fc)y|^_i_;,|(r)] |

(3.5)

At the origin, r = , of the polar coordinates (r, a) continuity requires that
<^(0,a) , w[0,a) and n(0,Q:) be independent of a . From (3.1) we thus get that:

fk{0)=wk{0)=gk{0) = 0, /c = l,2,,

(3.6)

202

Note that a condition on lOo(O) is not obtained but lOo(O) = w{0,a) . The
conditions (2.6) at r = 1 applied to (3.1a,b) yield:

a) fk{l)=0, k = l,2,...

b) /i(l)=0, k = l,2,...

c) T/;;c(l)=0, A; = 0,1,2,... (3.7)

The formal consistency of "order" of the system and number of boundary
conditions seems to be off by one since all of the equations are second order and we
do not have two boundary conditions on tyo('") . This is easily remedied by noting
that the equation in (3.3) for k = can be reduced to a first order equation. To
do this we multiply by r and integrate over [0, r] . In evaluating at r = we use
(3.6) and the assumptions that:

Imi [r<(r)] = lim [rX('-)]=0.
The result is the first order equation:

Yr Mr) + 7 [^ Mr) " - Mr)] = Jr Y. "" -^-W «^-W " i "^ ' (3-8)

n=l

The analytical problem is thus reduced to solving (3.2) for /c > 1 , (3.3) for k>l,
(3.4) for A; > 1 and (3.8) subject to the boundary conditions (3.6) and (3.7).

4. Numerical Procedures

To solve or rather to approximate the solution of the problem formulated in
Section 3 we first truncate the Fourier expansions, we then use difference approxima-
tions on the resulting system of O.D.E.s and finally we solve the nonlinear difference
equations by means of Newton's method and continuation procedures. We describe
these techniques below.

203

A. Truncation of the Fourier Expansions

Under the assumption that the series in (3.1) converge sufficiently rapidly we
replace them by the finite trigonometric expansions obtained by setting

fk{r) = Wkir)=gk{r)=0, k>K. (4.1a)

When we use (4.1) in the equations (3.2)-(3.8) we obtain a system of ZK second
order and one first order ordinary differential equations for the ZK + 1 quantities:

!k[r) , gk[r) , l<k<K; Wk[r) , 0<k<K. (4.16)

there are 6jFC + 1 boundary conditions in (3.6) and (3.7) when we terminate those
relations at k = K . We seek to solve this two-point boundary value problem
numerically,

B. Diff"erence Approximations .

We place a uniform grid of points rj = jh , 0.<j<M + l with r^+i = 1 on
the interval < r < 1 . At each point of this grid we introduce approximations to
the coefficients in (4.1b) with the notation

fk{rs) = fk,o , Qkirj) = gkj , Wk{rj) = w^j
We employ the diff"erence operators, for any mesh function, say Uj :

Then the discrete or difference approximations to (3.2), (3.3) and (3.4) are taken
to be:

204

cr,

D^D^-

{k - 1)(^-2) ■

.2^

' +

/;=-lJ +

D+D- + —Do - -^

rj

^l

fk,o

+ ■

er,

JD+D_-

(fc + l)(fc + 2)

Jfc+i.j - — ^ f?/:-!,; ~ 9k,j Y 9k+lj;

(4.2)

er.

D+D--

{k-l){k-2)

^J

t^;c-i,y +

-f

D+D--

{k + l){k + 2)

i

1 fc^

P+P_ + — -Do - -2 u^/:,y

Tj r J. J

(4.3)

(f)

• \ 2 r

Dj,D-

{k-2)[k-Z)

A

9k-2,: +

Hi

2 L'

2D+D- + — Do-

{k-l){2k-3)

rl

9k-ij

+

D+D- + — Do - -T

rp

+

e^r]

D+D-

2-2.,

••?

ffcj

+

J L

2B,I.- + liJ„-ii±iM±^

er.

(f

= Vi5--^* + '"'= + ''

'?

9;c+2,y

= -y- 'S';:-i,j + Sk+ij + ~Y Sk+i,i + Pk,3 + 2 ^'^•J" '
Each of these difference equations is imposed for

(4.4)

y = i,2,...,M,

7c = l,2,...,Jr .
The quantities R^j , S^^j , P;:,y , and Q^j are the obvious finite difference

approximations to the quantities in (3.5) centered at Tj . Since only first

derivatives occur in these expressions we employ Dqw^j to approximate

iy|j(ry) , etc. The remaining first order equation (3.8) is centered at the points

Ty-i = (y — 2)^ as follows:

D-Wqj + ■

„..,,__£_ (-i^-^)

/n,j + Jn,j-l\ C^n,] + ^^nj-]

D

-r,_. -, (4.5)

s 2

205

for

y = l,2,...,M + l .

The boundary conditions (3.6) and (3.7a,c) go over into the corresponding
conditions:

a) fk,o = Wk,o = gk,o = , k = l,2,...,K ;

b) fk,M+i = '^kM+i = ^ ^ k = l,2,...,K ; tyo,Af+i = . (4.6)

The remaining conditions, in (3.7b), are imposed in order to retain second order
accuracy as:

n f - /^.■M'+2 - fk,M n t 1 o T^

■^0 Jk,M+i = TTi = 0, k = 1,2,...,K .

Of course the meshpoint r^+s is not in [0,1] and so the values fk,M+'2 seem
extraneous. However they are eliminated by imposing the difference equations in
(4.2) at j = M +1 . The result, after using (4.6b) and the above, is for e = :

2
9k,M+^. = -J-^ fkM y k = l,2,...,K . (4.7)

For e > we must add the terms:

- [9k-l,M+l + 9k+lM+l + D+D- [fk-iM + fk+lM)]

The numerical problem is to solve the nonlinear system of difference equations
in (4.2), (4.3), (4.4), (4.5) and (4.7). These form 2KM + K + M-^l equations.
There are precisely that many unknowns {fk,j , Wk,j , gk,j } when the quantities
in (4.6) are eliminated. We go further and use (4.7) to eliminate the K quantities
{9kM+\} • Then we have only {2K + 1)M + 1 equations and unknowns.

206

C. Newton^s Method and Continuation .

To solve the difference equations we use Newton's method combined with con-
tinuation procedures to insure good initial. estimates of the solution as the param-
eters are varied. To do this efficiently the unknowns must be ordered in a manner
that simplifies the structure of the Jacobian matrix. To describe our ordering we
first introduce the vectors / . , g _ and w . of dimensions JK, K and K + 1 ,
respectively, by:

/y = (•'"i.J' -^s.y, • • . , /ic,y) , 1 < i < M ;

gj. = (^i,y, P2,; , . . . , 9Kj) , 1 < i < M -f 1 ;

^ J = (^o,i, wij, . . . , WK,j) , 1 < y < M . (4.8)

w

Recall that (4.7) gives: g = -r^ /,, (for the case e = 0) and so g can

be eliminated. The remaining (SiC -f l)M -M unknowns are represented in the
vector X defined by:

x^ = Ko;/f, £f, €,:--:f^. £^, ufL) • (4.9)

Now we order the equations in a corresponding manner. That is for a fixed j-
value (:.e. meshpoint) we take (4.5) and all of (4.2), (4.3) and (4.4) for \<k<K .
The equations ordered in this manner for j = 1,2,...,M and finally (4.5) for
J = M -f 1 can be written as a vector equation in the form

G(X;D,e) = 0. (4.10)

Here G has (SJf + l) components, each being one of the difference equations. We
have indicated the dependence of these equations on the parameters D and e as they
play a special role in the continuation procedures. For a fixed value of D and e we

207

denote a solution of (4.10) by X = X{D, e) . When D = and e = an
exact solution of the continuous problem is given by Poiseuille flow. Thus we easily
get a solution of the discrete problem in this case. As D or e deviates from
zero we can use the Poiseuille flow as an initial estimate of the discrete solution in
Newton's method applied to the system (4.10). This gives a sequence of iterates
{X^''^D,e)} defined by:

a) X^°^ (£>, c) = initial estimate ,

b) G-^iX^'');D,e) [x(^+'^ - ^(^^] = -G(X("); A^), ^ = 0,1,2,. .. .(4.11)

Here G v- is the Jacobian matrix which as a result of the above indicated ordering
has the block-band structure indicated below. Each square block is a matrix of
order (3Jf + l) x [ZK + 1) . There are M

5x

D

N ^ \

\

\

\

N

V

such rows of blocks. This array of blocks is bordered by one row and column cis
shown. All other elements in Gy are zero. Most of the computing eifort goes into

208

solving the linear algebraic systems in (4.11b). Thus to reduce the number of times
this must be done we seek accurate initial estimates.

One way to obtain good initial estimates is to use two lerms in a Taylor ex-
pansion of the solution with respect to changes in the parameter D , say. Thus we
use:

X(°) {D + 8D,€)= X{D, e) + SDX^ [D, e) (4.12a)

To obtain Xj^ we note, from (4.10), that it satisfies;

G^{X{D,e);D,e)X^ = -G^{X{D,e);D,,) (4.126)

This system is similar to those in (4.11b). In fact when Newton's method has con-
verged, the last time we solve (4.11b) we can also solve (4.12b) and thus X [D, e) is
determined with little extra work (i.e. only the backsolves and evaluation of
Gjj need be done). Continuation with respect to c can be done in an exactly
similar manner.

The method described in (4.11), (4.12) is known as Euler-Newton continuation.
It is extremely effective and usually converges quadratically. There are many refine-
ments regarding step length procedures, efficient solution of the block-banded linear
systems, approximation of Jacobians, etc., which we do not discuss here. Failure of
the method to converge does occur, however, and it usually signals the presence of
a bifurcation or fold point on the solution path (or family) being generated. Such
points or solutions are called singular because the Jacobian matrix evaluated at
these solutions is singular. Almost all such singular points are what we call simple
folds or limit points. In particular a simple fold with respect to D is a singular
solution, say [X^, Dq, cq], which has the properties that:

209

a) dimiV(G^) = 1 ; (i.e., all solutions of

G%-d) = are (j) = a 6 ; a € 5R , some 4>^ i^ 0)

(4.13)
6) G%^^{G\) {i.e. (G'^,V') 7^0 for all solutions of

Here G^. = G^{X^;Do,eo) and gj, = G^{X^;Do,eo) . All of the singular
solutions we have found in this work have been such simple fold points. We have
sought bifurcation points but have found none.

It is not difficult to circumvent the convergence problems near fold points. We
do this by using pseudo-arclength continuation as introduced in Keller (1977), That
is, we do not parametrize the solution path or family by D (as we assume has been
done above) but rather introduce a new parajneter 5 and a new scalar constraint
and seek to solve the inflated or augmented system:

a) G{X{s],D{s),e)=0

b) N{Xi3),D{s),s) = (^X{so), [X{s) - X{so)fj

+Diso) \d{s) - D{so)\ + (s - 5o) = (4.14)

Here [X(5o)> jC('So)] is a previously computed solution for e fixed in the present

dX

discussion and for s = sq . By X = -^ and D = -^ we denote the components of
a tangent vector to the solution path ^X{s),D{s)'j. The constraint (4.14b) simply
requires that the point [X(5),jD(5)] lie on the plane normal to this tangent at a
distance {s — sq) from the point of tangency.

We use the scheme (4.14) when the previous Euler-Newton scheme begins to
show signs of failure {i.e. too many iterations till convergence). We solve (4.14) by
Newton's method. The Jacobian of this system is

210

d[X,D) \Nx Nd I ^^•^^-'

This Jacobian is nonsingular at regular solution points and at simple fold points.
That is why our method has no difficulties in computing solution paths through
folds. To solve for the Newton iterates we use the Bordering Algorithm described
in Keller (1977) which is designed for systems with coefficients as in (4.15).

By differentiating in (4.14a) with respect to s we find that [X{s),i){s)] , the
tangent to the solution path, satisfies:

Qx^i^) + Qo^i^) = (4.16a)

To solve this we first solve

and then set

X{s) = D{s) ^{s) (4.16c)

However since the scale of 5 has not been determined we choose it to represent
(local) arclength along the solution path. Thus we require that

{X{s),X{s)) + D^s) = l
and using (4.16c) in the above we get

i?(5) = ±(y/l-<^,^>) ' (4.16cf)

The sign here is chosen so that {X{s),X{so)) > which determines the orientation
along the solution path.

211

We determine a new tangent only after having solved (4.14). Then we replace
[X(5o),I)(5o)] by the new tangent [X(s),jD(s)] and proceed as before.

5. Results of Calculations

In addition to the stream function and axial flow velocity we have computed
Re , the Reynolds number based on the mean axial velocity:

Re = 2\/5 / WQ{r)rdr ;
and the friction ratio (ratio of curved, ^c > to straight, 7« , wall friction):

We have computed solution paths with D varying for the following sets of
values of Fourier truncation, K , mesh spacing, h , and coiling ratio, £:

J. JC = 10 , /i = — •, £ = ;
40

IJ. ilT = 10 , h=—; e = , •£ = 0.1 ;

60

III. K = 20 , ^ = ^ ; e = , e = 0.1 , e = 0.2
Starting from the trivial state with u = T; = ty = Ofore = and D = we used
continuation with D increasing zs described in Section 4. In each of the three
cases a simple fold was found and arclength continuation was used to accurately
locate the fold and to traverse it. The solution branches were then continued with
decreasing D and, in each cajse, another fold was found. Again these folds were
located accurately and traversed to obtain a third branch in each of the three cases,
now with D increasing. For cases I and II, extensions of these third branches
continued well beyond where we could trust the numerical results. However for
case III a third and fourth fold were found, leading to five branches of solutions.
In Table 1 we list the critical value of the Dean number. Dm , at the m-th fold.

212

For cases II and III the fold solutions found for e = were continued in e up to
0.1 and for case III the continuation went up to e = 0.2 . These results are also
given in Table 1.

We call the family of solutions varying continuously with D in Dm-i < D <
D,n the "m-th branch" (Dq = 0). Our calculations seem to suggest that the
analytic problem has infinitely many branches although we have computed only
five of them. Graphs of ')c/ls vs D are given for cases I and III in Figures 2 and
3, respectively. On the first branch, that emanating from D = , the solutions
are of the classic form described by Dean — we call these "two-vortex" flows (see
Figure 4). These two- vortex flows persist on the entire first branch and over most
of the second branch down (in D values) to about D w 5000 where four-vortex
solutions gradually appear. These four-vortex flows are formed in the calculations
by the development, as D decreases on the second branch, of a small weak pair of
vortices about the axis of symmetry near the outer edge of the tube. This vortex
pair grows as D decreases and persists onto the third branch as D then increases
(see Figure 5). The four-vortex flows remain on the entire third branch and onto the
fourth branch down to I? w 14, 000 where six-vortex flows appear. We believe that,
as this process continues, 2n-vortex flows can form for all n = 1, 2, . . . . Indeed on
the fifth branch we have computed 8-vortex solutions at JD « 25,000 (see Figure
9).

In Table 2 we compare our computed values of qc/ls on the first branch with
various values reported in the literature (for two-vortex flows). The agreement is
quite good. Dennis and Ng (1982) have also obtained four-vortex solutions over
957.5 < D < 5000 . We claim that these solutions are on the third branch. They
were obtained accidentally in Dennis and Ng (1982) as a result of convergence
difirculties with increasing D values near 5000 . Then as D was decreased the
solution "jumped" back onto the first branch. This is typical of the behavior to be

213

expected near folds if no special technique for traversing them is used. Thus the
intermediate second branch was not obtained in Dennis and Ng (1982). In Table
3 we compare the values of the four-vortex solutions obtained in Dennis and Ng
(1982) with our values on the third branch. The agreement leaves no doubt as to
the identity of the two results. The somewhat larger discrepancies at D = 5000 is
due, we believe, to inaccuracies in Dennis and Ng where convergence difficulties
occurred. Graphs of the stream function and axial velocitj^ contour lines on the
third branch also agree well with those in Dennis and Ng.

Over the interval D4 < D < D3 we have obtained five solution branches. To
give some idea of how the solutions change we show in Figures 4-8 plots of contour
lines of the stream function and axial velocity for the solution with D = 8000 on
each of the five branches. In addition we display in Figure 9 the results for D =
25,000 on the fifth branch. The contour lines in each figure are at levels that diff'er
by one tenth the value between maximum and minimum values of the quantity
plotted. The values of these maxima and minima are given with each figure. The
small closed contours (or almost points) near the maxima or minima are at the
levels of 0.995 or 1.005 , respectively, of the critical values.

Least squares fits of the "^c/ls vs D curves with e = have been made in
the form

It
On branches m = 1, 2 and 5 we get the coefficient values:

ai = 0.3 , as = 0.25 , 05 = 0.15 and 61 = 62 = ^5 = 1/8 .

Other exponents have been used but the 1/3 power seems to fit the data best.
It is not clear, in light of the multiplicity of solutions and the unsettled nature
of the solutions for large D , what the significance of "asymptotic solutions" for

214

D -+ oo can be. Thus we do not address this problem here but merely present the
above fits for whatever use they may be.

During the course of this work we have benefitted from conversations with
Prof. A. Acrivos. We also wish to thank Prof. S.C.R. Dennis who first brought the
matter of multiple solutions to our attention and suggested that .we work on it.

215

References

Collins, W,M. (fc Dennis, S.C.R. " 1975 The steady motion of a viscous fluid in
a curved tube. Q. J. Mech. Appl. Math. 28, 133-156.

Dean, W.R. 1927 Note on the motion of fluid in a curved pipe. Phil. Mag.
4, 208-223.

Dean, W.R. 1928 The stream-line motion of fluid in a curved pipe. Phil.
Mag. 5, 673-695.

Dennis, S.C.R. 1980 Calculation of the steady flow through a curved tube
using a new finite-difi"erence method. J. Fluid Mech. 99, 449-467.

Dennis, S.C.R. <fc Ng, M. 1982 Dual solutions for steady laminar flow through
a curved tube. Q. J. Mech. Appl. Math. 35, 305-324.

Keller, H.B. 1977 Numerical solutions of bifurcation and nonlinear eigen-
value problems. In: Applications of Bifurcation Theory [ed. Rabinowitz),
pp. 359-384. Academic Press.

Lentini, M. 6i Keller, H.B. 1980 The Karman swirling flows. SIAM J. Appl.
Math. 38, 52-64.

Van Dyke, M.D. 1978 Extended Stokes series: laminar flow through a loosely
coiled pipe. J. Fluid Mech. 86, 129-145.

Winters, K.H. k Brindlej', R.G.G. 1984 Multiple solutions for laminar flow
in helicaJly-coiled tubes. AERE-R 11373, U.K. Atomic Energy Authority,
Harwell.

Winters, K.H. 1984 A bifurcation study of laminar flow in a curved tube of
rectangular cross-section. TP-1104, U.K. ARE, Harwell.

216

Table and Figure Captions

Table 1. Critical Dean number, Dm , at'the m-th fold in the solution branches.

Table 2. Comparison of 7c/7s on the two-vortex solutions of various works with the
present solutions on the first branch.

Table 3. Comparison of the four-vortex solutions of Dennis and Ng (1982) with the
present solutions on the third branch.

Figure 1. The tube cross-sections showing coordinates, velocity components, axial flow
distribution sketch and cross-flow streamlines sketch.

Figure 2. Friction ratio, 7c /7s i "s. Dean number, D , for case I: K = 10,
h = 1/40, e = .

Figure 3. Friction ratio, 7c/7s , vs. Dean number, D , for case III: K = 20,
h = 1/60, e = .

Figure 4. Axial velocity, «; , and stream function, (f> , contour lines: D = 8000,
K = 20, h = 1/60, € = . First branch: Max w = 0, Min «; = ,
Max 4> = 23.986, Min </> = .

Figure 5. Same as in Fig. 4. Second Branch: Max w = 625.956, Min u; = ,
Max (f) = 23.497, Min (/> = .

Figure 6. Same as in Fig. 4. Third Branch: Max w = 594.777, Min u^ = ,
Max (p = 22.962, Min (j) = -12.897 .

217

Figure 7. Same as in Fig. 4. Fourth Branch: Max w = 613.697, Min «; = ,
Max ^ = 21.783, Min (p = -8.7ld

Figure 8. Same as in Fig. 4. Fifth Branch: Max w = 622.831, Min u; = ,
Max (f) = 20.679, Min </) = -4.676 .

Figure 9. Axial velocity, w , and stream function, , contour lines: D = 25, 000,
K = 20, h = 1/60, 6 = 0. Fifth branch: Max w = 1412.730, Min w = ,
Max <l> = 31.494, Min (j) = -14.335 .

218

Table 1

K

h

e

Dl

^2

D3

D,

I.
II.

III.

10
10
20

1
40

1
60

1
60

12,120
12,752
25,146

951
950
955

15,642

7,725

II.
III.

10
20

1
60

1
60

0.1
0.1

19,963
27,508

1,130
1,138

18,179

10,576

III.

20

1
60

0.2

30,071

1,358

20,440

14,007

Table 2

D

Collins &
Dennis '75

Dennis &
Ng '82

Dennis '80

This Work

1000

1.550

1.548

1.546

1.548

2000

1.852

1.847

1.848

3000

2.064

2.063

2.065

4000

2.237

2.237

2.238

5000

2.392

2.377

2.383

2.383

Table 3

^c/'^s

w^(0)

Re

D

Dennis
& Ng'82

This Pfork

Dennis
& Ng'82

This Work

Dennis
& Ng'82

This VJork

2000
3000
4000
5000

1.8329
2.0463
2.2177
2.3662

1.8338
2.0472
2.2172
2.3527

1.0803
1.0514
1.0390
1.0332

1.0795
1.0522
1.0389
1.0368

192.9
259.2
318.8
373.5

192.8
259.1
318.9
375.7

219

to

tsJ

O

Figure 1.

^c

'/n

3.0

2.0

1.0

950

5000

10000

12752 15000

20000

D

Figure 2.

NJ

to

^c/.

rs

3.0

2.0

1.0

D3= 15643

D,= 25146

5000

10000

15000

20000

D

Figure 3.

AXIAL VELOCITY CONTOURS

STREAM FUNCTION CONTOURS

Figure 4.

223

AXIAL VELOCITY CONTOURS

STREAM FUNCTION CONTOURS

Figure 5,

224

AXIAL VELOCITY CONTOURS

STREAM FUNCTION CONTOURS

Figure 6,

225

AXIAL VELOCITY CONTOURS

STREAM FUNCTION CONTOURS

Figure 7.

226

AXIAL VELOCITY CONTOURS

STREAM FUNCTION CONTOURS

Figure 8.

227

AXIAL VELOCITY CONTOURS

STREAM FUNCTION CONTOURS

Figure 9.

228

Calculations of the Stability of Some Axisymmetric Flows
Proposed as a Model of Vortex Breakdown.

Ncssan Mac Giolla Mhutrit

Institute for Computer Applications in Science and Engineering,
Mail Stop 132C, NASA Langley Research Center,
Hampton, Virginia 23665, USA.

ABSTRACT

The term "vortex breakdown" refers to the abrupt and drastic changes of
structure that can sometimes occur in swirling flows. It has been conjectured that
the "bubble" type of breakdown can be viewed as an axisymmetric wave travel-
ling upstream in a primarily columnar vortex flow. In this scenario the wave's
upstream progress is impeded only when it reaches a critical amplitude and it
loses stability to some non-axisymmetric disturbance. We will investigate the sta-
bility of some axisymmetric wavy flows, which model vortex breakdown, to three
dimensional disturbances viewing the amplitude of the wave as a by"urcation
parameter. We will also look at the stability of a set of related, columnar vortex
flows which are constructed by taking the two dimensional flow at a single axial
location and extending it throughout the domain without' variation. The method
of our investigation will be to expand the perturbation velocity in a series of diver-
gence free vectors which ensures that the continuity equation for the incompressi-
ble fluid is satisfied exactly by the computed velocity field. Projections of the sta-
bility equation onto the space of inviscid vector fields eliminates the pressure term
from the equation and reduces the differential eigen problem to a generalized
matrix eigen problem. Results are presented both for the one dimensional, colum-
nar vortex flows and also for the wavy "bubble" flows.

229

1. Introduction: Vortex Breakdown.

The term "vortex breakdown" refers to the abrupt and drastic changes of structure that can
sometimes occur in vortex flows. Observations by Peckham & Atkinson [1957] of breakdowns
occurring in the leading edge vortex formed above a swept back lifting surface and a number of
studies demonstrating the serious aerodynamic consequences of such events (the slopes of the lift,
drag and moment curves are all altered by breakdown) stimulated early interest in the subject.
Since that time the literature on vortex breakdown has burgeoned. The interested reader is
referred to review articles by Hall [1972] and Leibovich [1978, 1984] for summaries both of the
experimental observations that have been made and also of the theories that have been proposed
to explain them.

Experimental observations are most easily made on vortex flows confined to tubes and the
bulk of the available data is for such cases. In one apparatus, used by a number of researchers,
water is passed radially inward through a set of guidevanes imparting swirl to the fluid which
then enters axially into a test section (a frustrum of a cone of very small cone angle), by means of
an annular channel formed between a bellmouth opening on the section and a centerbody whose
tip is aligned with the cone axis. The boundary layer shed from the tip of the centerbody forms a
well defined vortex core along the axis of the test section and dye injected through the tip allows
for flow visualization.

With this type af apparatus two parameters are within the easy control of the experimen-
talist, namely the amount of swirl imparted to the inlet flow and the volume flow rate through
the tube (eflFectively the Reynolds number of the flow). As the Reynolds number is increased for
a sufficently large, fixed value of swirl the breakdown assumes one of two characteristic forms.

230

Both of these are characterized by a rapid deceleration of the axial velocity component, occurring
in the axial distance on the order of one vortex core diameter, followed by the formation of a
stagnation point and (in some frame of reference) a region of reversed flow along the axis. The
two forms are easily distinguished in flow visualization studies as in one form, the spiral or S type
breakdown, the tracer dye assumes a spiral shape rotating in the same sense as the inlet fluid,
while in the other form, the bubble or B type breakdown, the dye assumes a form that looks
much like a body of revolution placed in the fluid. Our interest will be in this latter form of
breakdown which is sketched in Figure 1. Here we show "ideal" or averaged stream surfaces on
which the fluid particles travel in helical paths. (Leibovich [1978]).

Faler &; Leibovich [1977], Garg & Leibovich [1979] and the author [unpublished studies]
have used the non-invasive techniques of laser doppler anemometry to measure the velocity fields
both upstream and downstream of breakdown events. Outside a thin boundary layer along the
tube wall the experimental data is well fitted by the analytic profiles,

V{r) = }q(i - e-"'j (1.1)

W(r) = W^ + W^e-"'' (1.2)

W and V being respectively the axial and azimuthal velocity components while W^, W2, Q and a
are all constants (representative values are given by Garg & Leibovich [1979]). The profiles apply
to the downstream flow only in the mean, as the flow there fluctuates with time.

Upstream of the recirculation zone the flow is axisymmetric and steady. After breakdown
the vortex core expands to two or three times its upstream size and the constant W2 in the mean
axial velocity profile which had been positive upstream (jetlike flow) becomes negative (wakelike
flow). Downstream, within a few vortex core diameters of the breakdown, a turbulent wake is

231

invariably established . This transition to turbulence "switched on" by the coherent breakdown
structure provides a further incentive for its study.

Possibly motivated by the fact that the flows upstream of breakdown can be made to have a
high degree of axial symmetry, most of the research to date assumes that axially symmetric
processes are the important ones in vortex breakdown. As only axisymmetric disturbances can
cause a change in the axial velocity component as measured on the axis and as a deceleration of
this component is so pronounced in breakdown, it is clear that such disturbances play an impor-
tant role. Nevertheless, all transitions occurring in vortex flows as documented by Faler [1976]
are nonaxisymmetric and the flow within the bubble itself is unsteady with regular low frequency
oscillations. Furthermore, the stagnation point that defines the start of the recirculation zone is
not entirely fixed but wanders over a short range of the axis in a seemingly random fashion.

Leibovich [1984] proposed the following plausible scenario for the bubble type breakdown.
A finite axisymmetric disturbance, triggered off downstream, moves upstream in a columnar flow
that is nearly critical in the sense of Benjamin [1962]. (A supercritical flow, in this classification,
allows for the upstream propagation of infinitesimal axisymmetric waves while a subcritical flow
does not). Flows of the form (1.1,2) can indeed support axisymmetric dispersive waves and these
can propagate upstream in some situations (Leibovich [1970], Randall & Leibovich [1973]). Mov-
ing in this direction, the cross sectional area of the tube decreases causing the wave to amplify
and speed up. The conjecture is that, upon reaching some critical amplitude, these waves lose
stability to a non-axisymmetric disturbance. The growth of the asymmetric mode at the expense
of the axisymmetric wave, drains energy from it and this causes the wave to equilibrate at some
axial location in the diverging tube, much as is seen in experiments.

232

Our aim is to study the stability of some inviscid, wavy axisymmetric flows to three dimen-
sional disturbances with the amplitude of the waves viewed as a bifurcation parameter. We start
with a columnar flow in cylinderical coordinates of the form (0, VQ{r), H^oC''))' (^-g- (1-1)2))- (For
arbitrary C^ functions, Vq and Wq, all such flows satisfy Euler's equations). We then seek
axisymmetric wavy perturbations to this flow which satisfy the equations of motion, at least
approximately, for small amplitude. In terms of a stream function, rp and a circulation, k
(Lamb [1932]) Leibovich [1972] found solutions to Eulers equations of the form,

0(r,x) = Mr) + ^<f>{r)A{x) + 0{e% (1.3)

K{r,x) = /co(r) + e^{r)A{x), + 0{e% (1.4)

where z is a moving coordinate,

X = z - dt (1.5)

and d is a constant that must found in the calculation. The velocity components are given by

« = ---l-V', (1.6)

r az

V =

-«, (1.7)

rv = ^-l-rP. (1.8)

r ar

The columnar base flow is represented by tpQ^r) and 'Co(r).

The amplitude function, A{z,t) is governed by a Korteweg de Vries equation which has both
infinite and finite period solutions. The multiple scales analysis that was used to obtain these
solutions is strictly valid only for long period waves which are also the most interesting solutions
from a physical point of view. When doing the stability analysis we will confine our attention,
for numerical refisons, to solutions of the finite period, 2H and these are given exactly in terms of

233

cnoidal functions (Whitham [1974]). The structure functions, <f>{r), 7(r) and the wave speed d
are determined (numerically) from a second order, ordinary differential eigenvalue proble

lem.

For certain base columnar profiles d is negative and consequently the axisymetric wave pro-
pagates upstream. Figure 2 is a plot of the streamlines (1.3) in a frame moving with the wave for
such a case. The base columnar profile used here and throughout this paper is a purely swirling
flow; WQ{r) = and a = 14 in the notation of (1.1). The structure function (f) has been normal-
ized so that Max ^ = 1 and for this flow a recirculation zone (bubble) first appears in the stream-
line plot when the amplitude parameter, e reaches a value of 0.0155. For the value of e used here
the plot is clearly reminiscent of the bubble type breakdown.

Our aim is to study the stability of the flows (1.3,4) to three dimensional disturbances view-
ing e as a bifurcation parameter. The analysis will be carried out in a frame moving with the
wave, i.e. using the coordinates (r, x, 6). As the base flow is dependent on both the radial and
axial variables, r and x, the stability equations separate only in the azimuthal variable, 0. It will
be in our interest also to study the stability of a related columnar flow that is constructed by tak-
ing the two dimensional flow (1.3,4) at a single axial station, a: = 0, and extending it throughout
the cylindrical domain without variation. For obvious reasons we will refer to this flow as the
"mid-bubble" columnar flow and it is given explicitly as follows,

n(r) = V,{r) + i.7(r), (1.9)

^^iCO = W^o(r) + jr{r). (1.10)

Plots of these profiles for various values of e are given in Figures 3 and 4. Provided the wavy
flow (1.3,4) varies only slowly along the axis (as it will do if the period, 2^of ^(z) is very large),
we can look on the midbubble profiles as models for the full two dimensional flow. We conjecture

234

that the stability of these columnar flows (the equations for which separate in both x and 0)
should also be indicative of the stability properties of the full two dimensional flow.

In the rest of this paper we describe the numerical scheme used to solve the stability equa-
tions, we discuss its implementation and verification on the computer and finally we give results
obtained for the stability of the midbubble columnar and the axisymmetric wavy flows presented
above.

2. Numerical Methods.

For incompressible fluids the physical law of mass conservation reduces to the constraint that the
velocity vector of the fluid be divergence free. The pressure is not then a thermodynamic variable
determined by an equation of state but rather can be thought of as a Lagrange multiplier adjust-
ing itself instantaneously to ensure that this kinematical constraint on the velocity vector is met.
There is no evolution equation for the pressure nor does it satisfy any predetermined boundary or
initial conditions.

Numericists, seeking to solve the governing equations approximately, have found that their
greatest difficulty lies in the treatment of the pressure variable. While many ingenious methods
have been devised to overcome the difficulties, the treatment advocated in this work is in a
mathematical sense the most natural and off'ers many computational advantages. Here, the pres-
sure term is eliminated from the equations entirely and the divergence free condition is satisfied
exactly by the numerically obtained approximation to the velocity vector. Moreover, as the com-
ponents of the velocity are expanded in terms of series of polynomials that arise as the solution to
a singular Sturm Liouville problem, whose excellent approximation properties are well

235

documented (e.g. Gottleib & Orszag [1977], Quarteroni [1983]) convergence of our approximation
will be bound only by the smoothness of the solution and by the number of terms used in the
component expansions. For infinitely differentiable velocity fields we should expect to achieve
"exponential convergence" (Canuto et al.)

The essence of the method (originally due to Leonard & Wray [1982]) involves expanding
the velocity in a series of divergence free vector fields each of which satisfy the same boundary
conditions as the velocity. The infinite sums are truncated and substituted into the governing
equations, which are the Navier-Stokes or Euler equations linearized about the appropriate base
flow. Inner products are taken with vectors fields which satisfy inviscid boundary conditions.
This eliminates the pressure term from the equations and reduces the differential eigenvalue prob-
lem to a matrix eigenvalue problem. The eigenvalues determine the stability of the base flow and
the eigenfunctions are the set of coefficents in the expansions of the corresponding perturbation
velocity fields.

To examine how this method works we recall that it is well known (Ladyshenskaya [1966])
that L {D), the space of square integrable vector functions defined on a bounded domain
D {D C iZ"' n = 2,3) can be decomposed into those that are divergence free and whose normal
components vanish on the boundary and those that can be expressed as the gradient of a
diff'erentiable function defined on D. For this paper we will consider vector fields, defined on the
section of a cylinder T, which are periodic in both the axial and azimuthal variables, having as
their axial period the tube length, 2H.

We will decompose jD^(r) as follows.

L\T) = J{T) + /(T), (2.1)

where,

236

J{T) =

«ei2(r) (6) « -n = on ar,

(c) U| 5^ = U| 5^ .

(2.2)

Si and Sj represent the ends of the cylinder. Given in this form J( T) is clearly the space of

(a) incompressible, (b) inviscid, (c) periodic velocity fields.
The set of "viscous" velocity fields on T is a subset of J{T) denoted J^{T).

-^{T) = I H e J{T) I « = on ar [ .

An alternative representation of J{T) (Richtmyer [1978]) is given by,

(o) <u, VP> = for all p e C°°(f)

J{T) =

Vl^L\T)

(6) u\ s^ = «| 5,

where T is the closure of T and <*,•> represents the usual inner product in L^{T),

(2.3)

(2.4)

<u,t;> = r u-t^ rdrdOdx

(2.5)

The space J{T) endowed with this inner product is a Hilbert space and a closed subspace of
L\T). The projection of ^^(r) onto J{T) will be denoted by 11. It is clear that vectors of the
form VP are perpendicular to all u in J{T) and in fact (Ladyshenskaya [1966]),

j\t) =\ ucL\T) I u=yp for some pin C\T)\.

n then has the following properties.

n : L\T) ^ j(r),

(2.6)

(2.7)

237

n u = u for all u e J{T), (2.8)

n VP = for all p € C\f). (2.9)

To determine the linear stability of a flow U to say, viscous disturbances which are periodic
in X and we consider whether infinitesimal perturbations to U grow in time. Therefore we
linearize the Navier-Stokes equations about Uand seek solutions in J^{T) of the form,

u(r,a:,^)e-''". (2.10)

The character of a then determines the linear temporal stability of U. If o- = a + i^ where a, /?
are real then,

/9>0 => U is unstable,

/9 = => U 13 neutrally stable,. (2.11)

/?<0 => U is stable

The equations that must be solved have the form,

to-u = ^'u + Re'^Su. (2.12)

E and S are operators defined on J( T) as follows,

^u = -n(vxw) (2.13)

and

Eu = Tl{u}xU + nxu), (2.14)

where fi and w are respectively the base and perturbation vorticities, {U=\/xU, cj = \jxu).

Some suitable nondimensionalization has introduced the Reynolds number,

Re=-^-2., (2.15)

Rq and Uq being characteristic length and velocity scales associated with the base flow and u is
238

the kinematic viscosity of the fluid. We can take Rq to be the radius of the tube and Uq to be the
maximum value of the columnar base flow azimuthal velocity, VQ{r).

The VP term in the Navier-Stokes equation has been eliminated by projection onto J{T).
Projection of the stability equation onto a finite dimensional subspace, Jff{T) of J{T) is achieved
in practice by taking the inner product of the equation with basis vectors for J^{T). This pro-
cess eliminates the operator 11 from the equation, for given any vector / in L^{T) and any vector
A in J{T) we have that,

<n/, A> = </, A>, (2.16)

as projections are self adjoint and as the projection of any vector in J{T) is itself.

It is worth emphasising that even when we are solving the viscous stability equations, we
still project the governing equations onto the space of inviscid vector fields. The reason for this is
that having found a velocity u such that the vector / defined as,

/ = tVu — upiU - flxu + z/yxw (2'17)

is orthogonal to all A in J{ T) then / e •/ ( T) and so there exists a scalar function p (a pressure)

with f_~^- If, however, / were in J° (T), which contains J (T) then the existence of a pres-
sure is not guaranteed and consequently u may not correspond to a physical solution.

Leonard and Wray [1982] demonstrated a divergence free vector function expansion for
viscous velocity fields, defined on a cylindrical domain that are Fourier decomposable in both the
axial and azimuthal variables. We will construct a somewhat different set of basis vectors here.

239

The velocity field, u, satisfies the continuity equation and is Fourier decomposable in x and
^, which means in effect that only two of its three components, u, u, w are independent. This
motivates the introduction of two vector families, x^ in an expansion of the form,

H = S I «2n*m2f;('-) + «2„-UmX;(r)[ e •(*' + "•^) (2.18)

nkm\ J

The components of the vectors x^ are found zis follows. Expand two of the velocity components,
say the first and the third, independently as.

« = E«2„-u,„/„-(r)e*■^*'■^'"''^

nkm

(2.19)

^ = S«2„*m/;(r) «•■('- + '"''),

(2.20)

where /^(r) are complete sets of polynomials chosen to satisfy the boundary conditions that are
imposed on u, w. The r and x components of x^ have now been picked and it remains for us to
chose the components in a manner that ensures the vectors x^ e '^** "^ '"^^ are divergence free.
Consider for example, X^(r).

V-(x;(r)e'(*« + '"^)) = 0, (2.21)

=>

{rf-{r))' + irnx-,e = 0, (2.22)

where the prime denotes a derivative with respect to r. This equation gives us the 6 component of
X^(r). Rescaling, it is found that an expansion of the form (2.14) is possible for non-zero azimu-
thal wavenumbers where,

X;^ = (w„-(r),-(r/„-(r))',o), (2.23)

240

X^{r) = (0,-rA/+(r),m/„+(r)) (2.24)

and such an expansion will guarantee that u is divergence free. This expansion is clearly incom-
plete for azimuthal wave number zero, (m = 0), i.e. for axisymmetric flows. For that case the
following expansion vectors can be used.

x;(r) =

ikfnir), 0,-^(r/„-(r))'

(2.25)

X^ir) = (o, /;(r),o) (2.26)

The polynomials /*(r) must be chosen so that the vector u given by (2.14) satisfies
appropriate (viscous or inviscid) boundary conditions on the walls of the domain, T and is single
valued at the origin, r = 0. We will denote the polynomials used in the inviscid case by a^{r)
reserving /^(r) for viscous expansions. We have then upon truncating (2.14) an approximation
to u of the form,

N K M

VjiKM = S I! S '^nkmDj,km{r,X,e) (2.27)

n=l k=-Km=-M

where.

D^km{r,x,e) = ^(r;*,m)e •■(*' + '»''). (2.28)

The projection vectors will have the same form as the expansion vectors, i.e. we will project
with vectors, ^p^{r,x,0) , where

^P,('->=^>^) = |f(r; A, m)e •■('- + "') (2.29)

for

/ = l,...,iV; p = -K,...,K; q = -M,...,M
with the vectors ^ being given by equations (2.23 - 26) using the inviscid polynomials, a^{r) for

241

the components.

It is possible to choose the polynomials a[^{r) and /^(r) in many different ways. Leonard &
Wray [1982] in their consideration of certain turbulence simulations employed an unusual set of
Jacobi polynomials to reduce the bandwidth of the final matrix system. These polynomials were
also used by Spalart [1983] in his simulation of boundary-layer transition. Moser & Moin [1984]
in their work on the infinite Taylor Couette system, used Tchebychev polynomials and incor-
porated the weight function, against which these polynomials are orthogonal, into the projection
vectors. Here, we will construct the basis vectors from Legendre polynomials. All of the above
sets are solutions to singular Sturm Liouville problems and consequently we can expect expan-
sions in terms of any of these polynomials to exhibit excellent convergence properties.

The single valuedness criterion, which must be applied along the centre line of the tube for
the vector u, causes the polynomials /*(r) and fl*{r) to depend on m, the azimuthal
wavenumber (Joseph [1970]). One appropriate choice for af'{r) is,

a^r) = rP,(2r - 1) f„^ ^^ ^^

«;"(»•) = (1 - r)P,{2r - 1) if I m| = 1, (2.30)

a-{r) = r(l - r)P,(2r - 1) if 1 "^l ^ h

where the radial variable has been scaled by the tube radius and P/(r) is the Legendre polynomial
of order I (Abramowitz & Stegun [1970]). The corresponding choice for /^(r) is,

f:{r) = r(l - r)P„(2r - 1) f„, ^n ^^

/„"(0 = (1 - rfP„{2r - 1) if I m| = 1, (2.31)

/„-(r) = r(l - r)V„(2r - 1) ^^1^ ^ ^'

242

As all of our stability problems separate in the azimuthal direction, this dependence on m
presents no difficulty. We solve separate problems for each azimuthal wavenumber; so having
chosen an m the expansion and projection sets are fixed throughout the calculation . Indeed, in
theory there is no difficulty even if the problem at hand is truly three dimensional; however some
care is required in implementing the method to ensure that the correct radial polynomial set is
being used for each azimuthal component of the velocity.

3. Ixaplexnentation and Verification of the Method.

Equation (2.12) is solved approximately by using the expansion u^^Jif ^°^ H ^^^ taking inner pro-
ducts of the equation with the projection vectors, Ay^, to get a generalized matrix eigen problem
for the eigenvalues a and the eigenvectors a (the coefficents in the expansion Uj^km)' This matrix
problem can be written as,

ffAa

^^±°

(3.1)

The matrix A is purely real and arises from the fact that the expansion and projection vectors
are not orthonormal.

The Kronecker delta symbol, ^,y arises because the Fourier bases employed in the axial and
azimuthal directions are orthogonal. The matrix B arising from the convection terms is also
purely real.

^Mp*,m = <^„, {W^km)^lL + ^^D^km>- (3-3)

Finally, the matrix C arising from the viscous terms is purely imaginary.

243

Clnpk,m = <^p,,V>^VxD^km>- (3-4)

Using the orthogonality of the Fourier bases it can be written as,

^Inpkqm = ^In^pk^qm. (3-5)

The form of the matrix B depends on the base flow U. For columnar flows which are independent
of X and it is possible to find a matrix B such that,

^Inpkjm = B^Jpk^qm. (3.6)

The stability of these flows can be determined by solving the 0{N) generalized matrix eigen
problem,

crAa = B + -^C a. (3.7)

•"^ir.^

On the other hand, for the axisymmetric wavy base flow (1.3,4) we have

Blnpkqm = ■^MyJt'^^pt (3.8)

and the stability equation is the 0\ [2K + 1)xN\ matrix equation.

cAa

^^±^

(3.9)

where A and C are the 01 {2K + 1)xN\ block diagonal matrices A^^S^^ and Ci^S^^ respectively.
The matrices depend parameterically on the wavenumbers of the projection and expansion vec-
tors so we separate them into submatrices that can be evaluated independently of these and the
other parameters (in particular e) occurring in the base flow. The submatrices are evaluated once
and then stored in the computer. The required integrations can be done at very little cost by util-
izing the orthogonality properties of the expansion and projection polynomials. The full matrices
are then be reassembled without the need for doing any further integrations.

244

One can always band the A and C matrices by appropriate choice of the polynomials /^(r)
and af'{r). However the matrix B will generally be full, though for certain rather simple base
flows such as the Hagen Pouiseille flow considered be Leonard & Wray [1982] it is also possible to
band B. The matrix A was inverted to produce a regular eigenvalue problem in place of (3.1)
and the QR algorithm was used to extract the eigenvalues. We also note that the matrix problem
we get when considering the inviscid stability of base flows is a purely real one and consequently
the eigenvalues occur, as they should do, in conjugate pairs.

A computer code has been written which implements the method we have been describing to
solve the stability problems, both viscous and inviscid, for all columnar flows and for axisym-
metric flows of the form (1.3,4). Both the direct and adjoint versions of the stability problems
can be solved. The adjoint viscous stability problem is to find a u in J^{T) such that

iXu = ^'e + -^-^l- (3-10)

The operator E is the adjoint operator to E and is given by,

E'u = -n( nxu + Vx(«xCO) (3.11)

The direct and adjoint spectra obtained by solving (2.12) and (3.10) should, of course, be conju-
gate to each other and how well a numerical scheme reproduces this theoretical result is a test of
its accuracy.

We verified the code by calculating the stability of rotating Poiseuille flow,

C7 = ( 0, V^r, W^{1 - r2) j (3.12)

Cotton et al. [1980] found that this flow with V^ = 0.2147 and Wi = 1.0 was neutrally stable to
disturbances having azimuthal wavenumber, m = 1 and axial wavenumber, k = —1 for a

245

Reynolds number of 156. The following table lists the most unstable eigenvalue we found for this
flow with the same wavenumber pair for the disturbance. The first column of the table gives N,
the number of radial basis vectors that were used to obtain the eigenvalue given in the next two
columns, N is also the order of the matrix problem that needs to be solved at each step.

Most unstable eigenvalue found for the rotating Poiseuille flow (3.12)
with m = 1, k = -1, Vi = 0.2147, Re = 156.0.

iV

frequency

growth rate

4
6
10
14
18
22

-0.00029

-0.00279

-0.00284

-0.002847

-0.002847898

-0.002847898

.00334

.00101

.000001

.0000001

.0000001379

.0000001378

The convergence is exponential in N or some power of N and there is no evidence of significant
roundoff" error. The following table lists the corresponding eigenvalue found by solving the
adjoint viscous stability problem for the same wavenumber pair and baseflow.

Eigenvalue found by doing the adjoint viscous stability problem for the flow (3.12),
with m = 1, k = -1, Vi = 0.2147, Re = 156.0.

N

frequency

growth rate

4
6
10
14
18
22

-0.00270

-0.00280

-0.00284

-0.002847

-0.002847898

-0.002847898

-.000094

-.000013

-.000004

-.0000002

-.0000001379

-.0000001379

Clearly the agreement between the adjoint and direct results is excellent. Inviscid stability
results for flows of the form (3.12), obtained using our code also compare well with results in the

246

literature. These results instill confidence in the accuracy of the numerical method and in the
code that implements it, at least for the case of columnar flows.

4. Stability Results for the Vortex Breakdown Model Flows.

In this section we will present the results obtained to date for the stability of the midbubble
columnar, (1.9,10) and the wavy vortex (1.3,4) flows. Although these flows are inviscid we will
consider their stability to both viscous and inviscid disturbances (i.e. we will solve the linearized
Euler and the linearized Navier-Stokes equations for these flows). The justification for doing a
viscous analysis is that the "real" flow is of course, viscous. Moreover, the inclusion of the higher
order dissipative terms eliminates certain technical difficulties that arise due to the existence of
critical layers in the neutrally stable eigenfunctions for columnar flows (Drazin & Reid [1981]).

We will begin by presenting the viscous results for the midbubble columnar flows. We found
that thirty radial vector modes {N = 30) were adequate to resolve the most unstable eigenmode
(i.e. the mode whose eigenvalue had the largest imaginary part) to three decimal places for these
flows at low Reynolds numbers and that this number increeised as the Reynolds number grew.
Frequent checks were carried out on the accuracy of the computed eigenvalues both by increasing
the order of the expansion and also by computing the adjoint spectrum for the same set of flow
parameters. The diff'erence between the most unstable eigenvalue as computed by the direct and
adjoint versions of the code was always less than 1%.

Having fixed the number of radial expansion vectors in our system the viscous eigenvalues
for the midbubble flows depend on four parameters,

<r = a{m,k,e,Re). (4.1)

247

The base columnar flow (e = 0) was found to be stable to all disturbances. It seems that even for
very small values of c (values for which there is no recirculation zone in the full two dimensional
flow) the midbubble flows are unstable. This is documented in the following table which gives
bracketing values for the critical Reynolds number for various values of e.

Bracketing values for the critical Reynolds number.

Various values of e and m = — 1.

e

Stable for Re

Unstable for Re

0.000

Stable for all Re

0.005

550

600

0.010

180

200

0.015

160

180

0.020

110

120

0.025

60

80

0.030

42

45

For large enough values of e disturbances having both positive and negative azimuthal
wavenumbers can destabilize the midbubble flows with the negative azimuthal modes giving rise
in general to the largest values for the growth rates. In particular disturbances with azimuthal
wavernumber, m = -1 were found to be the most dangerous. This is shown in the following
table.

248

Bracketing values for the critical Reynolds number.
Various values of m with e = .03.

m

Stable for Re

Unstable for Re

-1
-2
-3
-4
-5

42

90

700

1200

1400

45

100

800

1400

1600

For fixed values of m and e, a two parameter {k, Re) study was carried out. With e = .03
and m = -1 we obtain the stability diagram shown in Figure 5. The stability boundary appears
to be a parabolic curve which is markedly asymmetric with respect to the k = line. Within this
curve the base flow is unstable to a range of axial wavenumbers; however, there is a "tongue" of
stable wavenumbers that gradually thins out as the Reynolds number is increased. The k = 1
mode is the final one to be excited, this does not happen until Re = 4500 (approximately).

We now consider the stability of the midbubble flows to inviscid disturbances. The invis-
cisid stability of columnar flows is governed by an ordinary differential equation, the Howard-
Gupta [1962] equation. A number of analytic results obtained from this equation exist in the
literature. Leibovich & Stewartson [1982] showed that a sufiicent condition for the instability of a
columnar flow is that the function.

Fir) = F(r)A'(r)(A'(r)r(r) + W'{rf] (4.2)

be negative somewhere in the domain of interest. (A is the angular velocity, — V and V is the cir-

r

culation rV.)

Ih9

Th function F{r) is eeisily evaluated for the midbubble profiles and this has been done for a
range of values of e. It was found that F first becomes negative only when a critical value of e is
reached, e = 0.132. Consequently the midbubble flows are guaranteed to be unstable by the
Leibovich-Stewartson criterion for values of the parameter e slightly below those needed to pro-
duce a recirculation region in the full two dimensional flow (recall this happens for e = 0.155).

A normal mode stability analysis was carried using the divergence free expansion method
and the numerically obtained results confirm and somewhat extend the predictions of the
Leibovich-Stewartson criterion. Once again we found that the disturbances giving rise to the
largest growth rates had azimuthal wavenumbers, \m\ =1. Figure 6 shows how the maximum
growth rate varies (almost linearly) with the amplitude parameter, e. In the figure we can see
that the normal mode analysis extends the previously obtained results in that midbubble flows
with e < 0.0132 are also found to be unstable, even though F{r) > for these flows. The max-
imum growth rates also increase with the size of the axial wavenumber, | A;| , as shown in Figure
7.

We conclude that the midbubble flows are definitely unstable on both viscous and inviscid
grounds for values of the parameter, e below those needed to produce a reversed flow region in the
full two dimensional wavy flow. Moreover the most destabilizing disturbances have azimuthal
wavenumbers, | m| = 1 and short axial period, (| k\ > 1). The unstable inviscid eigenfunctions
tended to have regions of steep gradient near the origin and this made their resolution difficult.

While these midbubble stability results we have just reported tend to support the conjec-
tures made about vortex breakdown, they are not encouraging for the numericist seeking to inves-
tigate the stability of the cnoidal wave flows (1.3,4). As we noted earlier, the stability equation

250

for these flows do not separate in x and consequently we have to solve 0{N{2K+1)) matrix
eigenvalue problems for each flow. It would seem to be necessary to include modes having a short
axial period in our expansion, Vjfjcj^ which means that K will be large. The inclusion of these
modes is also dictated by physical considerations. We should like the disturbance to include
modes that scale with the dimensions of the bubble and in fact visualization studies indicate that
the asymmetric unstable modes do have short axial periods. However, our experience with the
midbubble flows show that these unstable modes are difficult to resolve radially, consequently the
number of radial vector functions, N in our expansion must also be large.

With these constraints the normal mode analysis of the cnoidal wave flows becomes prohibi-
tively expensive, (reacall that the number of operations needed to extract all the eigenvalues of a
matrix is proportional to the cube of its order). To alleviate the cost problems most of the runs
for these flows were done with the axial period of the cnoidal wave fixed at 1 (JT = 0.5, units are
tube radii). While such short period flows violate the assumptions needed to produce the solu-
tions (1.3,4) it W£is hoped that these flows would be unstable to disturbances with smaller values
for K. Indeed convergence studies indicate that adequate resolution in the axial direction was
obtained with if « 10. Up to 40 radial modes were used in the study, leading to an 0(840)
matrix eigenvalue problem when K = 10. (The calculations were performed on an Floating Point
Systems 164 series vector processor at Cornell University, using a vectorized version of the QR
algorithm, optimized for this machine and using some 16 megabytes of memory.) The adjoint
problem Wcis also solved on each run and only those eigenvalues which agreed well between the
adjoint and direct calculations were considered.

Unfortunately the short period flows behave less like the midbubble flows and more like the
underlying, stable columnar flows. This is indicated in Figure 8 which plots the least stable

251

growth rates found for the various values of e as the Reynolds number is increased (m = — 1 in
this plot). All these short period cnoidal wave flows are stable, though marginally so. The dis-
turbances with I m| = 1 are again the most unstable. Figure 9 shows the least stable growth rates
found for some other azimuthal wavenumbers. The inviscid stability runs that were performed
also failed to turn up any definite evidence of instability for these flows.

We have considered the viscous and inviscid stability of some axisymmetric flows which are
said to model the bubble type of vortex breakdown. The investigation was carried out by
expanding the perturbation velocity in terms of the new set of divergence free vectors presented
in section 2. The stability results for the midbubble flows support the conjectured mechanism for
breakdown and it was found that the most dangerous disturbances have a short axial period and
azimuthal wavenumbers, \ m\ =1. The two dimensional flows we considered all had rather short
axial periods and these do not model the physical phenomena well. No conclusions as to the sta-
bility of these flows can be drawn because the normal mode analysis used here can only prove ins-
tability (by finding a growing disturbance). No evidence of instability was found but this may
well be because we failed to include enough axial modes in our expansion for the disturbance.
The study clearly points out that linear stability investigations for complex base flows are far
from trivial from a computational point of view.

5. Acknowledgements

The author is happy to acknowledge the assistance of Professors Philip Holmes and Sidney
Leibovich. Computations were carried out on equipment at Cornell University.

252

6. Bibliography

(1) Abramowitz, M. and Stegun, I., eds., 1970, Handbook of Mathematical Functions with For-
mulas, Graphs and Mathematical Tables. 10th ed. U.S. Gov. Printing Office.

(2) Benjamin, T.B., 1962. The theory of the vortex breakdown phenomenon. J. Fluid Mech. 14,
593.

(3) Canute, C, Hussaini, M.Y., Quarteroni, A. and Zang, T.A. (To appear). Spectral Methods
with Applications to Fluid Dynamics. Springer- Verlag, New York.

(4) Drazin, P. and Reid, W., 1981. Hydrodynamic Stability. Cambridge University Press.

(5) Faler, J.H. and Leibovich, S., 1977. Disrupted states of vortex flow and vortex breakdown.
Phys. Fluids 20, 1385.

(6) Garg, A.K. and Leibovich, S., 1979. Spectral characteristics of vortex flow fields. Phys.
Fluids 22, 2053.

(7) Hall, M.G., 1972. Vortex breakdown. Ann Rev. Fluid Mech. 4, 195.

(8) Howard, L.N. and Gupta, A.S., 1962. On the hydrodynamic and hydromagnetic stability of
swirling flows. J. Fluid Mech. 14, 589.

(9) Gottlieb, D. and Orszag, S., 1977. Numerical Analysis of Spectral Methods: Theory and
Applications. CBMS-NSF Regional Conference Series on Applied Mathematics, Vol. 26.
SIAM, Philedelphia.

(10) Joseph, D.D., 1970. Stability of Fluid Motions: Volume I. Springer Tracts in Natural Philo-
sophy Vol. 27, Springer- Verlag, New York.

(11) Ladyshenskaya, O. A., 1969. The Mathematical Theory of Viscous Incompressible Flow.
Gordon and Breach, New York.

(12) Leibovich, S., 1970. Weakly nonlinear waves in rotating fluids, J. Fluid Mech. 42, 803.

(13) Leibovich, S., 1978. The structure of vortex breakdown. Ann. Rev. Fluid Mech. 10,221.

(14) Leibovich, S., 1984. Vortex stability and breakdown: Survey and extension. AIAA J. 22,
1192.

(15) Leibovich, S. and Stewartson, K., 1983. A sufficent condition for the instability of columnar
vortices. J. Fluid Mech. 126, 335

253

(16) Leonard, A. and Wray, A., 1982. A new numerical method for simulation of three dimen-
sional flow in a pipe. Proc. International Conference on Numerical Methods in Fluid Dynam-
ics, 8th, Aachen. Lecture Notes in Physics, Vol. 170 (ed. E. Krause). Sprincer-Verlae New
York, pp. 335-342. J ^ ^ B,

(17) Mac GioUa Mhuiris, N., 1986. Numerical Calculations of the Stability of Some Axisymmetric
Flows Proposed as a Model for Vortex Breakdown. Ph.D. Dissertation, Cornell Univ.

(18) Moser, R.D. and Moin, P., 1984. Direct Numerical Simulation of Curved Turbulent Channel
Flow. NASA TM 85974.

(19) Peckham, D. and Atkinson, S.A., 1957. Preliminary results of low speed wind tunnel tests
on a Gothic wing of aspect ratio 1.0. Aeronaut. Res. Counc. CP 508.

(20) Quarteroni, A., 1983. Theoretical motivations underlying spectral methods. Proc. Meeting
INRIA - Novosibirsk, Paris.

19

(21) Richtmyer, R.D., 1978. Principles of Advanced Mathematical Physics: Volume I. Texts and
Monographs in Physics, Springer- Verlag, New York.

(22) Randall, J.D. and Leibovich, S., 1973. The critical state: a trapped wave model of vortex
breakdown. J. Fluid Mech. 53, 495.

(23) Salwen, H., Cotton, F.W. and Grosch, C.E., 1980. Linear stability of Poiseuille flow in a
circular pipe. J. Fluid Mech. 98, 273.

(24) Spalart, P.R., 1983. Numerical simulation of boundary layer transition. Proc. Interna-
tional Conference in Fluid Dynamics, 9th. Lecture Notes in Physics, Vol. 218 (ed. H.
Akari). Springer- Verlag, New York, pp 531-535.

254

Figure 1. Axisymmetric bubble type vortex breaJcdown (after Leibovich (1978]).

255

, ,, 1 1 1 1 I I 1 I I I I I 1 I I I » »»' 'I '''''''''' ' "^

[tit;

^ » 1 1 1 1 I ' ' *

Figure 2. Streamlines (1.3), f = .02, H ~ r.

256

o

a>

>

.2
<

-1.

-1.
-1.

,3

,2

1

1

2

3

4

,5

,6

7

8

,9

1

^

^

/i

- //
:/ /

- 1

- 1

7

.1

.2

£ = (underlying columnar flow)

£ = 0.01

£ = 0.02

£ = 0.03

1

.3

.4

.5
R

.6

.7

.8 .9

1.0

Figure 3. Axial velocity profiles for the midbubble flow.

00

O

CD

>

•♦-»

3

N

Figure 4. Azimuthal velocities for the xnidbuble flows.

E

c

>

CD
X

<

Reynolds number, Re

Figure 5. Stability diagram for the midbubble flow with e = .03 and m = — 1.

259

to
o

2.8
2.6
2.4
2.2

2.0
■g 1.8
f 1.6

o

^ 1.4

1.2

=3

E

I 1.0

.8
.6
.4
.2

o4

1

1

1

1

1

1

1

J

1.2E-02 1.4E-02 1.6E-02 1.8E-02 2.0E-02 2.2E-02 2.4E-02 2.6E-02 2.8E-02 3.0E-02

£

Figure 6. Maximum growth rates plotted against e, m = — 1 and A: = 6.

ro

o

o

^.8

2.6
2.4

—

O £ =
A £ =
V £ =

0.30
0.25
0.20

o

o

2.2

-

o

2.0
1.8
1.6

-

A

o

A

o

A

A

A

1.4
1.2

^"

V

V

o

A

^\

A

V

V

V

1.0
.8
.6

—

V

o

§

A
O

o

V

.4

—

O V

.2

—

1

1

1

1

1

A

T 1

1

1

1

1

1

-6 -5 -4 -3 -2

-10 12
Axial wavenumber, K

Figure 7. Maximum growth rates plotted against axial wavenumber, (inviscid analysis for
the midbubble flow), m = — 1.

ts3

ON

1.00E-02r

a>

CD

■1.00 E- 02

-2.00E-02
-3.00E-02
-4.00E-02

JC _l

o

CD

5.00 E- 02

-6.00E-02

-7.00E-02

-8.00 E- 02

■9.00 E- 02

-O.lOE-02
-O.llE-02

1

1

1

1

1

A-
V-
X-

e
e

£
£

0.015
0.010
0.005

1

1

100 140 180

220 260 300 340 380 420 460 500 540 580
Reynolds number

620

Figure 8. Least stable growth rates found for the full two dimensional flow (1.3,4),
H= .h, m = -1.

o
o

-.1.
-.2

-.3

-.4

-.5

-.6

-.7

-.8

-.9

-1.0

-1.1.

-1.2

-1.3

-1.4

-1.5

-1.6
-1.7

100

V

^-X-

.-><^*

X--

m
m

/^

V m =

m =

1/

1

1

-2

-3

-4

-10

I

200

300 400 500

Reynolds number

600

700

to

Figure 9. Leeist stable growth rates found for various values of m, H = 0.5, m = — 1.

NUMERICAL STUDY OF VORTEX BREAKDOWN

M. Hafez

G. Kuruvila

Vigyan Research Associates, Inc.

and

M. D. Salas
NASA Langley Research Center

Abstract

The incompressible axisymmetric steady Navier-Stokes equations and the
Euler equations are solved numerically to model the breakdovra of a vortex.
The solutions obtained for the Euler equations show a "vortex breakdown-like"
structure, their behavior is very different from that of the Navier-Stokes
solution which are obtained at low Reynolds number. The details of the
numerical algorithms used are presented, and the results obtained are compared
to those in the literature at the same Reynolds number.

Research was supported by the National Aeronautics and Space
Administration for the first and second author under NASA Contract No. NASl-
17919.

264

1. INTRODUCTION

Under certain conditions, it has been observed that the vortex shed from
the highly swept leading edges of a delta wing can change its structure
abruptly. The change is characterized by either a spiral deformation of the
vortex axis or the formation of a stagnation point along the vortex axis
followed by a bubble of recirculating flow. Downstream of this structural
change, the flow appears to be highly sensitive to perturbations and is
usually turbulent. This sudden change in structure is known as vortex break-
down. The effect of vortex breakdown on the aerodynamics of a wing is very
important, since it degrades the performance of the wing and can set a limit
on the maximum attitude achievable by the wing. The phenomenon has been
studied, both experimentally and theoretically, for the last 30 years, but no
really satisfactory theory exists to explain it. The reader is referred to
the two reviews of the subject given by Leibovich [1,2],

Our interest in vortex breakdown was aroused by the claim of Hitzel and
Schmidt [3] that vortex breakdown could be predicted on the basis of the Euler
equations. We consider that the numerical studies of flow over a delta wing
by Hitzel and Schmidt are too superficial to warrant such a conclusion. The
flow over a delta wing at high angles of attack is too complex and requires
too many computational resources to allow an indepth study. How could the
problem be formulated such that it would lend itself to an investigation of
the relevance of the Euler equations vis-a-vis the Navier-Stokes equations? A
drastic simplification of the problem is. required. Fortunately, experi-
mentalists have already achieved this by studying vortex breakdown within the
confines of cylindrical tubes. In addition, two numerical investigations of
the Navier-Stokes equations have been presented [4,5] for this problem; in one

265

case [4] steady solutions were obtained, while in the other [5] the solutions
appeared unsteady. Perhaps an additional investigation could shed some light
onto this problem. The purpose of this work is, therefore, to investigate the
possibility of simulating vortex breakdown with the Euler equations, studying
the relation of these solutions to those of the Navier-Stokes equations and
comparing the latter to those in Refs. 4 and 5.

2. MATHEMATICAL FLOW MODELS

An incompressible steady axisymmetric flow with swirl can be described in
terms of a streamf unction, ;|); azimuthal vorticity component, u; and a
circulation, k. In cylindrical coordinates (x,r,e) the Navier-Stokes
equations are:

' r

"^ (r^). + '^'xx = ™

(u.), + (wo,)^ -H (£|)^ = \- (.„ + (1)^ + co^J

uK^ + WK^ = ^ (<^^ - F -^r + ^xx) (2.1)

where < = rv, ai = w^ - u^^, and R^ is the Reynolds number defined in terms
of the free-stream axial velocity, the vortex core radius, and the kinematic
viscosity of the flow. The velocity components in the x,r,e directions are
denoted by w, u, and v, respectively. In terms of the streamfunctlon, w
and u are given by:

266

w = —

(2.2)

Is

u = -^

The inviscid equations are obtained from Eqs. (2.1) by letting R •»■ ".
In the inviscid limit, it is clear that the circulation becomes constant along
a streamline. It can also be shown that the total enthalpy, h, becomes
constant along streamlines. Moreover, the vorticity component oi can be
related to the gradients of the circulation and the total enthalpy by:

2 dh 1 d(K^) ,, ,.

^" = ^ dlF - I-dT ^^'^^

where the total enthalpy is given by:

h = p +-^ (u2 + v2 + w2) (2.4)

and p is the pressure. Notice that in the absence of swirl (v = 0), oi/r
becomes constant along a streamline. We also notice that the contribution to
the vorticity (given by eqn. (2.3)) due to circulation term does not depend
on the sign of k but only on its magnitude. The functions K(\})(x,r)) and
h(\lj(x,r)) are determined in terms of the specified inflow profile K(T|)(o,r))
and h(T|>(o,r)) at the upstream boundary, provided that i|^(x,r) is positive
(i.e., outside a recirculation bubble). Inside the bubble, the k and h
distributions are not known. In fact, within the inviscid model a
discontinuity is admissible across the streamline forming the bubble (the
separating streamline). One way to avoid this problem is to invoke analytic

267

continuation of the functions <(<(;) and h(ij;) for negative i(). In the
present work, the dependence of k and h is known analytically for positive
ijj from the assumed initial profiles. The same functional dependence is
assumed for negative ip. As a side point, it should be mentioned that since
K vanishes along a separating streamline (k = on the axis) and k is
analytically continued inside the bubble, it is reasonable to assume that k
changes sign inside the bubble; and, as a consequence, the swirl velocity in
the bubble has the opposite sense to the swirl in the main flow. This is not
the behavior observed with the viscous problem at Re = 100 and 200.

In solving the Navier-Stokes equations, the viscous terms play an
important role in the neighborhood of the separating streamline by preventing
the formation of discontinuous solutions. A similar role is played by the
artificial viscosity terms in Euler calculations based on primitive variables
(i.e., velocity components). However, it may be argued that although the
artificial dissipation is critical in singling out a solution, the solution
may be independent of its form and magnitude. In the least square formulation
used in this work, there is no explicit or implicit artificial dissipation.
In fact, the truncation errors for the central differences to be used are of a
dispersive nature. It is, however, the assumption of analytic continuation
of K and h, for the inviscid problem, that rules out any discontinuities.

If we let the vortex core radius at the upstream boundary be r = 1 and
the radius at the farfield boundary be r = R, the inflow profiles at x =
are given by:

268

u(r) =0 < r < R

v(r) = Vr(2 - r^) r < 1

(2.5)
v(r) = V/r r > 1

w(r) =1 < r < R

where V is the maximum circumferential swirl velocity at the edge of the
vortex core. These profiles are the same as those used in Ref. 4. From these
profiles, it follows that the circulation at the upstream boundary is given
by:

K^ = lev^t^ (1 - Ti))2 ^ < 1/2

(2.6)

K^ = V^ ,|; > 1/2

and that the vorticlty component is given by;

2
rco = lev^ (1 + 2iJ)2 - 3^)[j- - ^) i, < 111

ro) = t > 1/2

(2.7)

2
In terms of a perturbation streamf unction ^ = £— - \|) , the equation governing

the inviscid flow is:

"^xx ■*■ ^ ('I'r/^^r = -^V^a^Y (2.8)

where

269

a^ = 4 (1 + 2ii)^ - 3i)) il) < 1/2

(2.9)
a^ = ^ > 1/2

We notice that since 'F vanishes at the axis and if we require "V to
vanish in the far field, then the trivial solution 4" = 0, corresponding to
cylindrical stream surfaces, is a solution of the above equation. We are,
however, interested in nontrivial solutions.

Using standard central difference approximation, equation (2.8) leads to
a nonpositive definite matrix which is difficult to solve by standard
relaxation schemes. To avoid this problem, a least-square variational
formulation is obtained for the function

F(^u',w') = //„((!l - w')' r + (l2£ + u')' r + (w; - u; + AV^ |i ^)2JdQ

(2.10)

where n extends over the domain of interest and u and w are the
perturbation velocities in terms of "F. The first term in parenthesis in the
kernel of Eq. (2.10) corresponds to the definition of w' , the second term
corresponds to the definition of u', and the last term corresponds to Eq.
(2.8). Each of these terms should vanish in the steady state. To form the
kernel of Eq. (2.8), each of the above terms is multiplied by an arbitrary,
but positive, weight function. The choice of r as the weight function for
the first two terms and the cube of unit length for the last term was made to
simplify the form of the resulting equations. From the function (2.10), the
following Euler equations are easily obtained [6]:

270

¥ m;^), - ^ - (w; - uj (i . i4^^

u -ru = ^ +w +2

(2.11)

where

w -rw=-1'+u -g
rr r xr °v

'+V a ,„
g = — z — ^

3. NUMERICAL FORMULATION

3.1 Invlscid Problem

Equations (2.11) are solved using a staggered grid for 1, u , and w'.
With "^ defined at i,j nodes, u' between nodes of horizontal lines, and
w' between nodes of vertical lines, the resulting discrete equations are:

[y

i+i,j " ^'^i.j "^ '^i-i,j^ ,

Ax

liii '^l.J+1 " \i '^i,j " '^i,j- l
Ar

^ ''i,j+l/2 ''i,j-l/2

4VV

= r

l.j

^ ^i,j+l/2 " ''i,j-l/2 _ "i+l/2,j " "i-l/2,J

Ar

Ax

1 +

4VV^

^ ^ ^

^"i+3/2.j " ^"i+l/2,j "^ "i-l/2,.1^

Ax

- r u = ^+1»J i»-1

2 ''i,j"l+l/2,j Ax

' ^ ^ ^

^ K-fl,j-H/2 - ^i-H,j-l/2 - ^.j+1/2 "■ ^i,j-l/2) , ^gi+l,j - gi,j)

AxAr Ax

271

Kj+3/2 - ^^,j+l/2 -^ ^i.j-1/2) ' 4'_.,-^_

Ar

2 ^i,j+l/2''i,j+l/2 ~ Ar

("1+1/2. i+1 " "1-1/2.1+1 " "1+1/2.1 + "1-1/2. j) _ ^^l.j+1 H,p . V

AxAr Ar v-J-J-^

where

X, . = (1-1) Ax 1 = 1,2. ..I
i,j max

r, . = (j-1) Ar j = 1,2. ..J

l,j -^ '' ' max

(3.2)

The first equation of (3.1) Is solved for Y by the Zebra vertical line over-
relaxation algorithm; the second equation Is solved for u by direct
Inversion of horizontal lines; the last equation is solved for w by direct
Inversion of vertical lines. For the first equation of (3.1), the boundary
condition consists of ^ = all around the domain. For the second equation,
boundary conditions are required at 1 = 1+1/2 and 1 = I^j^^^ - (1+1/2).
Boundary conditions for u at 1 = 1+1/2 are obtained by solving

t"x ~ ^''r ^ 2^] ~ t'^x "^ ''"'] " ° ^^-^^

at 1 = 2; while at 1 = 1^^^^^^ - (1+1/2), boundary conditions are obtained by
solving

t"x ~ ^^r "^ S)] + V^y, + ru'] = (3.4)

272

^^ ^ = Imax ~ ^* ^^^^ t^'^™ ^^ square brackets appearing in Eqs. (3.3) and
(3.4) vanishes in the steady state. The change in sign between Eqs. (3.3) and
(3.4) is introduced to add to the diagonal dominance of the discrete
equations. Similarly, for w the equation

[\ - (u^ - g)] + [\ - rw'] = (3.5)

is solved at j = 2 to obtain w at j = 1+1/2; while at j = J - 1 , the
equation

[Wj. - (u^ - g)] - ['F^ - rw] = (3.6)

is solved to obtain w' at i = J ^^ - (1+1/2).

3.2 Viscous Problem

The first equation of (2,1) is discretized using second-order-accurate
central-difference formulas. To the second and third equations of (2.1) the
time terms w^. and k^ are added to the left-hand side of each equation,
respectively. The convection terms of these two equations are discretized
using upwind first-order accurate formulas, while the diffusion terms are
discretized using second-order-accurate formulas. The time terras are
discretized using first-order backward derivatives. Unlike the inviscid
problem, a nonstaggered grid is used for the viscous problem. The discrete
equations are

273

'1+1

J -^^i,j ^'^i-i,j ^ ^ . '^i.j+i - ^ij _ !ij

Ax

Ar^ ^^1^+1/2 "^

±,j-l/2 ^'J ^'J

At 2 Ar 2 Ar

w. . - w. . 0) . , , . - to ,

■ i.j ' i,j' i.J i-l>j , i>j ' i>j' 1+1»J i».1
2 Ax 2 Ax

U. ,0),

2 2

'i.j"'l,j ^1+1, j " '"l-l.j

r 3

i,j 2Axr, .
i.J

Re

Ar2

10. . , , - (0. , , (J^ . t»)j ,1 J ~ 2(0, , + (0. , ,

i»j+l i»j-l _ i.J . i+l.j i.j i-l.j

2 2

2Ar r. . r, . Ax
i.j i.J

i.j ij . i.j ' i.j' i.J i.J-1 + i.j ' i.j' i.J+1 i.J
At 2 Ar 2 Ar

2 Ax 2 Ax

Re

"i. j+1 ^^UiJ ^ijLJ::! _ ^i.j+1 " ^i.J-1 ^ ^i+l.j

2k. . + k. 1 .
i.J i-l.j

Ar

2Ar r

i.j

Ax

(3.7)

Quantities with a bar are taken at the new time or iteration level. The time
step is chosen equal to Ax. (Note that the identity u + w = — was used
to simplify the convective terms of the second equation above.) The Eqs.
(3.7) are solved by vertical line over-relaxation with the following boundary
conditions:

274

At X = 0,

2

K = 4ViJj(l - i|)) < r < 1

K = V l<r<R

u = i|) /r ;

XX '

at r = 0,

rl) =

K =

0) = ;

at

r = R,

r

K = V

0) = ;

at the outflow, x = L,

ip =
^x

K^=

(0=0

X

To improve the convergence rate of the viscous problem, the acceleration
method described in Ref. 7 was used.

275

4. DISCUSSION OF RESULTS

In order to measure the deviation of a solution from the trivial
2
solution ill = -J—, we define the norm of the perturbation streamfunction as

(max max „ \l/2/

Tests were performed to determine the required number of mesh points and the
required locations of the farfield boundaries to achieve a certain level of
accuracy. Two of these tests are illustrated in Figures 1 and 2 for the
inviscid problem. Figure 1 shows the asymptotic behavior of the
streamfunction norm as the number of mesh points in the axial direction is
increased, holding all other parameters fixed. Figure 2 shows the effect of
the location of the outflow boundary on the norm. From this study, it was
concluded that for the inviscid problem a minimum spacing Ar = Ax = 1/16
was required and that R > 2, L > 4 was also required. The same
requirements were found for the viscous problem for 100 < Re < 200, except
that the location of the outflow boundary had to be increased to L > 10.
Figures 3 and 4 show that the same solution is obtained for the inviscid
problem with L = 5 and L = 10. For all cases presented, residuals were
driven to machine zero, 0(10" ).

A summary of the results is given in Figure 5. This figure shows the
norm defined by Eq. (4.1) as a function of the square of the swirl parameter
V. Two nontrivial branches were found for the inviscid problem. The first
branch, indicated in the figure by the closed circles, corresponds to
axisymmetric vortex breakdown-like solutions. Figs. (3), (4), and (6)
illustrate the streamline topologies found in this branch. The same branch

276

was found by Ta'asan [8] using a multigrid algorithm to solve Eqs. (2.8). As
shown In Figure 5, our results and those of Ta'asan are in good agreement. The
problem with this branch is that as the swirl parameter is increased, the size
of the bubble decreases. This behavior contradicts the experimental

observations. The second inviscid branch, indicated by the open circles,

2
intersects the first at approximately V = 0.575. For values of V near the

intersection of the two branches, the numerical algorithm developed a limit

cycle where a single bubble splits in two. The two bubbles later coalesce and

the cycle is repeated. The limit cycle prevented convergence to a steady

state. Ta'asan only encountered the first branch and was able to continue

this branch down to the axis, as shown in Figure 5. The streamlines

corresponding to the second branch are illustrated in Figure 7. Obviously,

this -branch is not of the vortex breakdown type. It is believed that the

second branch, although a solution to the least square problem, is not a

solution of the original inviscid problem (Eqs. 2.1 with Re ■»■ «). The

evidence for this comes from inserting the least-square solutions into the

original equations and evaluating the residuals. When this is done for the

2
first branch, residuals of the order of Ax are found. For the second

branch, the residuals are of order Ax, and remain at the same level when the

mesh is refined.

For the viscous problem, results are presented in Figure 5 for Reynolds

numbers 100 and 200. The axial velocities obtained here and those obtained by

Grabowski and Berger [A] are compared in Figure 8. The agreement is good when

we consider that Grabowski and Berger used a much coarser but highly stretched

mesh, slightly different outflow boundary conditions, and did not converge

their solutions to the same level as in this work. It also appears that the

277

results obtained by Krause et al. [5] are anomalous, since they were unable to
obtain steady-state solutions for the same cases studied here. Their failure
to reach a steady state could be a result of the outflow boundary being at
L = 5, too close to the inflow boundary. In our work, we found this location
for the outflow boundary to lead to a large open bubble (see Figure 9), but
the solution was steady nonetheless.

Figure 10 illustrates the changes in the bubble structure as the swirl
parameter is increased with Re = 100 held fixed. The same is illustrated in

Figure 11 for Re = 200. It is interesting to see the very rapid change in

2
the norm that occurs at Re = 200 and V « 1.27. (See Figure 5.) This

behavior opens some questions about possible hysteresis and bifurcation at

higher Reynolds number. However, our present approach is not capable of

handling much higher Reynolds numbers well; and, therefore, these questions

will be considered at a later time. Figure 12 shows the minimum value of the

axial velocity component on the axis as a function of V for Reynolds numbers

of 100 and 200. The point at which the recirculation bubble first appears

corresponds to the first intersection of these curves with w = 0. The second

intersection corresponds to the point at which the recirculation bubble lifts

off the axis.

5. CONCLUDING REMARKS

Numerical solutions of the Euler equations were obtained and a vortex
breakdown-like topology was observed. Those solutions were in good agreement
with those obtained by Ta'asan [8]. For the Navier-Stokes equations,
solutions were also obtained with vortex breakdown-like topology. These

278

latter solutions were in good agreement with the results reported in Ref. 4.
The behavior of the inviscid solutions with increasing swirl was not
consistent with the behavior of the Navier-Stokes solutions at low Reynolds
number nor with experimental observations. (Experimental results showing
bubble-type vortex breakdown are usually obtained at higher Reynolds numbers.)
A future study will investigate the high Reynolds number limit of the Navier-
Stokes equations and compare it to the Euler solutions obtained here.

279

References

[1] S. Leibovich, "Vortex Stability and Breakdown: Survey and Extension,"
AIAA J. , Vol. 22, No. 9, 1984, pp. 1192-1206.

[2] S. Leibovich, "The Structure of Vortex Breakdown," Ann. Rev. Fluid Mech .,
Vol. 10, 1978, pp. 221-246.

[3] S. M. Hitzel and W. Schmidt, "Slender Wings with Leading-Edge Vortex
Separation: A Challenge for Panel Methods and Euler Solvers," J.
Aircraft . Vol. 21, No. 10, 1984, pp. 751-759.

[4] W. J. Grabowski and S. A. Berger, "Solutions of the Navier-Stokes
Equations for Vortex Breakdown," J. Fluid Mech. , Vol. 75, Part 3, 1976,
pp. 525-544.

[5] E. Krause, E., X. G. Shi, and P. M. Hartwich, "Computation of Leading
Edge Vortices," AIAA Paper No. 83-1907, Computational Fluid Dynamics
Conference, Danvers, Massachusetts, 1983.

[6] R. Courant and F. John, Introduction to Calculus and Analysis , Vol. 2,
Chapter 7, pp. 737-768.

[7] M. Hafez, E. Parlette, and M. D. Salas , "Convergence Acceleration of
Iterative Solutions of Euler Equations for Transonic Flow
Computations," AIAA Paper 85-1641, AIAA 18th Fluid Dynamics and Plasma-
dynamics and Lasers Conference, July 16-18, 1985, Cincinnati, Ohio.

280

[8] S. Ta'asan, "A Multigrid Method for Vortex Breakdown Simulation," ICASE
Report to appear.

[9] Xun-Gang Shi, "Numerische Simulation Des Aufplatzens von Werbeln," Ph.D.
Thesis, September 1983, Technischen Hochschule Aachen, West Germany.

281

IMI

0.06 -

0.05 -

0.04

0.03

0.02
0.01

20 40 80
I max

160

Figure 1. Convergence of the norm of the inviscid streamf unction ^ with
increasing resolution in the axial direction, holding L = 5,
R = 2, V^ = 0.4, and Ar = 1/16 fixed.

0.032
0. 031 h

//xl,// 0.030

0.029
0.028

3 4 5

6 7 8
L

9 10

Figure 2. Effect of increasing the length of the domain on the norm of the
inviscid streamfunction 1", holding R = 2, Ar = Ax = 1/16.

282

Figure 3. Computed streamline pattern for V = 0.2, L = 5, R = 2, and Ax = Ar = 1/16

ro

00

Figure 4. Computed streamline pattern for V^ = 0.2, L = 10, R = 2, and a^ = A'^ '^ 1/16.

II ^11

0.12
0.10
0.08
0.06
0.04
0.02

/ ^^

hk
t^

%

-V

In viscid
1st branch A^

2nd branch O
Viscous

Re = 100 A

Re = 200 A

P .^.

J I I

0.4 0.8 1.2 1.6 2.0 2.4
V^

Figure 5. Norm of the streamf unction Y as a function of V^.

284

Figure 6. Computed streamline pattern for V^=0.5, L=5, R=2,
and Ax = Ar = 1/16.

Figure 7. Computed streamline pattern for V^ = 0.9, L = 5,
R = 2, and Ax = Ar = 1/16.

285

( a ) Re = 100
V=0.9

8 10

( b ) Re = 200
V=0.9

8 10

( c ) Re = 200
V=1.0

w

Figure 8. Comparison f velocity on vortex axis between present results
(solid line) and those of Ref. 4 (dashed line).

286

Figure 9. Computed streamline pattern for Re = 200, V = 0.8944, L = 5,

R = 2, and Ax = Ar = 1/16. The shortness of the domain results
in a large open bubble. This case corresponds to the same
conditions of Figure 4.14, Ref. 9.

287

(a) V = 0.9

^:z::>

(b) V = 1.0

(c) V = 1.2

(d) V = 1.3

(e) V = 1.5

Figure 10.

288

Computed streamline patterns for Re = 100, L = 10, R = 2,

Ax = Ar = 1/16, and increasing values of V. Details of the

bubble structure are shown on the insets.

(a) V = 1.0

(b) V = 1.1

(c) V = 1.12

(d) V = 1.15

Figure 11. Computed streamline patterns for Re = 200, L = 10, R = 2,

Ax = Ar = 1/16, and increasing values of V. Details of
bubble structure are shown on the insets.

the

289

O

• Re = 100
▲ Re = 200

0.1-

mm ur

-0.1

-0.2

1

1

1

1

1

J

0.9 1.0 1.1 1.2 1.3 1.4 1.5

V

Figure 12. Minimum velocity on the axis as a function of V for Re = 100 and 200.

MULTIGRID METHOD FOR A VORTEX BREAKDOWN SIMULATION

Shlomo Ta'asan
Institute for Computer Applications in Science and Engineering

ABSTRACT

In this paper we study an inviscid model for a steady axlsymmetrlc flow
with swirl. The governing equation is a nonlinear elliptic equation which has
more than one solution for a certain range of the swirl parameter. The
physically interesting solutions have closed streamlines that look like vortex
breakdown ("bubble"-like solutions). A multigrid method is used to find these
solutions. Using an FMG algorithm (nested iteration), the problem is solved
in just a few multigrid cycles.

Research was supported by the National Aeronautics and Space
Administration under NASA Contracts No. NASl-17070 and NASl-18107 while the
author was in residence at ICASE, NASA Langley Research Center, Hampton, VA
23665-5225.

291

1. INTRODUCTION

In this paper we study an inviscid model for steady axisymmetrlc flow with
swirl, which has solutions with closed streamlines. These solutions have a
structure similar to that observed experimentally as "bubble"-like solutions
when vortex breakdown occurs [4],

Using a streamfunction-vorticity formulation to the axisymmetrlc
incompressible Navier-Stokes equations, it was found [3] that one can reduce
the problem to a single nonlinear elliptic equation for the streamf unction, in
case of a special inflow flow and some regularity assumption on the vorticlty.
This nonlinear elliptic equation for the streamfunction has more than one
solution. The trivial, represents a uniform flow and is of no physical
interest. The other shows a "bubble"-like structure, the target of our
numerical study.

In solving the problem numerically, the problem is reformulated in terms
of a perturbed streamfunction, i.e., the deviation from the trivial solution.
In terms of this perturbed streamfunction, the trivial solution is represented
as an identically zero solution. Our goal then is to find non-zero solutions
which have "bubble"-like form.

The approach we have taken in finding these solutions is to seek first for
a bifurcation point from the trivial branch of solutions. By introducing a
continuation parameter, we can then start marching on a branch of non-trivial
solutions that bifurcate from that point. One choice of a continuation
parameter is arc length [1]. Another choice, which is simpler but may not be
good in general, is the norm of the perturbed streamfunction. The natural
parameter in the problem, a swirl velocity parameter, is not good enough since
it cannot "choose" the non-zero branch as can the former parameter. We

292

therefore choose the norm as a continuation parameter, making the swirl
velocity parameter an unknown to be determined by the solution.

The multlgrid approach used for solving the problem is similar to the one
used in [5] for solving the Bratu problem. The relaxation In this method
consists of three steps: (1) a local relaxation to smooth the error; (11) a
step to update the norm of the solution; and (ill) a step to update the swirl
velocity parameter. An FMG algorithm (nested iteration) is used. That is, a
solution for the prescribed norm is found first on the coarsest level, and
then interpolated to finer levels, where on each level a few basic multlgrid
V-cycles are performed before proceeding to yet finer level.

The coarsest level, when solved to get an initial approximation for finer
levels, uses a continuation method. Here the problem was solved first for a
small norm, and then the norm is gradually increased until the prescribed norm
is reached. Each time the norm is increased, the solution of the previous
step was used as initial approximation. By solving for a bifurcation point
from the trivial solution, a first approximation for the smallest norm problem
was obtained.

Once a solution on the coarsest level is obtained for a prescribed norm,
it is possible to solve finer grid problems without continuation.

The same problem we are discussing here was treated by a completely
different method and is reported in [3]. There, a single grid method was used
with a least squares formulation of the problem. The amount of work needed
for that approach is considerably larger than the one reported here. Computed
solutions by the two different formulations are in good agreement.

293

2. ON DERIVATION OF THE GOVERNING EQUATION

We summarize here the derivation of the equations used in the numerical
process as given in [3]. In cylindrical coordinates (x.r.O) the
incompressible Navier-Stokes equations can be written in terms of a stream-
function \li, vorticity to, and circulation k as

*r
'^ — r ■•■'i'xx^ ^'^ (2.1a)

(uo.)^ + (wo.)^ + _ =^L +i^^_i£_+ 1 (2.1b)
r X *- r -J

uk + wk = 4- I k - - k + k 1 ('2 Ic")

r X Re |_ rr r r xxj ^.-^.ic;

where k = rv, to = w^ - u^ and Re is the Reynolds number. The velocity
components in the x,r,9 directions are w, u, v, respectively, of which w
and u are given in terms of the streamfunction by

w = -^ (2.2a)

u = - -^ (2.2b)

It is shown in [3] that in the inviscid case (Re = »), one finds that the
circulation k and the vorticity to are functions of the streamfunction ij;
only. Therefore, k and to can be determined outside the "bubble" from the
inflow boundary condition. In the model discussed it is assumed that the same
functional dependence of k, to on ij; is true also inside the bubble

294

(negative ij)). This Imposes some regularity on the solution.
For the inflow conditions

v(0,r) =

w(0,r) = 1,

fVp r(2 - r^)

!Vo/r

r < 1
r > 1,

(2.3a)

(2.3b)

it is possible to write k and o) in terms of the streamfunction as

16 vj /(I - T(;)2

k^(0,r) =

i|) < 1/2
t > V2

(2.4a)

(16 vj(l + 2i|)^ - 3iJ<)(r^/2 - ij^)

w(0,r) =

i(^ < V2
4* > V2

(2.4b)

and therefore, the equation obtained for i|j is

rC'I'j./r)^ "^ '''xx = ~ ^^0 "^(*)(''' ' '^^Z^)

(2.5a)

where

a^(i|;) =

C4(l + 2rJ;^ - 3^)

ij; < V2
If) > V2 .

(25b)

The reduction of the governing equations to a single nonlinear elliptic
equation is possible if the relation ip = f(r) in the inflow boundary can be
inverted to get r = g(ii'). When g(i^) is introduced in the expression for

295

V at the inflow boundary, one has v as a function of \|) in that boundary

and therefore k(i|/), td(i|)). Note that, in general, one cannot expect to

analytically invert the relation ^ = f(r), and so the reduction of the

governing equations is possible only for very special inflows.

2
Numerical experiments were done in terms of <)> = 'J' - •?— , which is a

2 ^

perturbation from the trivial solution ^ = —- that represents a uniform flow.

3. NUMERICAL ALGORITHM

3.1. Discretization

2
The equation for ^ = \l) - r /2 is given by

r(7 <t>^)^ + *xx "^ ^ ^0 ""^^^^^ ° °' " " ^°'^^ "" ^°'^) (3.1a)

<t> = 0,

on 9n

(3.1b)

where

a^<|))

'4 ([. - 1 + ~ (24) - 1 + r^)

Equations (3.1) are discretized as

<|> + |_ < 1/2

otherwise

(3.1c)

i+l,j

h2 7

^j+1

+1

♦?P

V^/'*'ij "*i.j-i^

+ V^ «2(,|,J_.)<(.J_j = 0, in n^

(3.2a)

(|)J = 0, on 80^ (3.2b)

296

where fi = {nh,mh), < nh < a, < mh < b} .

3.2. General Strategy for Solving the Dlscretlzed Equations

Equation (3.2) has the trivial solution (J) = for any Vq. This
solution corresponds to a uniform flow and is not interesting physically. We
seek solutions which represent vortex breakdown so that lltj) II ^0, where

ll(t)^l|2 = h^ E (|)^.. (3.3)

Iterating on equation (3.2) by any iterative method may lead us to the trivial
solution. In order to rule out this possibility, we specify the norm of the
discrete solution we want to find, while making free the swirl velocity
parameter Vq.

To summarize, we solve equation (3.2) for ((}> , Vq) under the

constraint

l<f^ll^ = gQ, (3.A)

where gQ is given,

A relaxation scheme for ((}> ,Vq) in equation (3.2) together with the
constraint (3.4) is described next.

3.3. Relaxation

Equations (3.2), (3.4) form a nonlinear system of equations for ((j) ,Vq).
The relaxation used for this system has three steps: (i) a local process for

297

smoothing <{) in equation (3.2); (ii) a global change to satisfy (3.4); and
(iii) updating the swirl parameter Vg. That is, one relaxation consists of
doing (i), (ii), and (iii) successively,
(i) local relaxation

Scan the point (i,j) € f2 in lexicographic ordering; at each point
(i,j) solve (3.2) approximately for <)),. by applying one Newton
iteration.

(ii) global step

Compute B = / g„/lli])^ll^ .
Then make the change

(iii) updating Vq

Change Vq such that the following equation holds

<L^ *^ + 4V2 cx^ify^^, ^S = <f^A^> (3.5)

where L (}> is the discretization of Lij) = r(— (\, ) + (}> ,<•,•>

denotes the inner product, <u,v> = h J] u . , v, . , and f"^ is the

ij
right-hand side of equation (3.2). (In a multigrid process f" is

nonzero on coarse grids.)

We now come to the description of the multigrid algorithm used to solve (3.2),
(3.4) for (<t,^, Vq).

298

3.4.1. Basic Cycle ;

Given a sequence of discretizations with mesh sizes
h > h2 >•••> h , where hj^ = 2hj^^j. The hj^-grid equation is generally

written as

L^ ^^ = f^ (3.6)

where L^ approximates L*^"*"^ (k < m) (e.g., they all are finite-difference

approximations to the same differential operator). The algorithm for

~k
improving a given approximate solution (J) to (3.6) is denoted by

t- ^ MG(k, ?^, f'") (3.7)

and is defined recursively as follows:

If k = 1, solve (3.6) by several relaxation sweeps; qtherwise do steps
(A) - (D):

(A) Perform v, relaxation sweeps on (3.6), resulting in a new
approximation (j) .

(B) Starting with ^ =1 <}) , perform one cycle

<|) ^ MG(_k-l, (() ,L ^ +1, (f-L<)))J.

(C) Calculate

<^ - <^ + \_i[i> - \ * J.

(D) Perform v^ additional relaxation sweeps on (3.6) starting with
1^ and yielding the final ?^ of (3.7).

299

k- 1 — k- 1
In this algorithm I , I are fine-to-coarse grid transfer operators;

Ij^_2 is an interpolation operator. We refer to the above cycle as MG(v,,v„).

In the notation of this section (3.6) includes both equations (3.2) and (3.4).

The basic cycle described above is for improving a given approximation on

level k. The full multigrid (FMG) process involves solving the problem on

the coarsest grid, interpolating it to finer grids, and making the cycle

MG(Vj^,V2) a few times after each refinement.

3.4.2. Full Multigrid Algorithm (FMG)

1. Solve (3.6) for k = 1, using a continuation method (see remark
below).

2. Set k = k + 1 and

~k k ~k— 1 k

<|) = n, _. (^ , where n, _. is a bicubic interpolation.

3. Perform Y(k) times the cycle
?^ ^ MG[k, ^^, f^).

4. If k < m, go to step 2; otherwise stop.

A Remark on Step 1 of the FMG Algorithm (Continuation Method)

Since the problem involved is a nonlinear one, and we are using a Newton
iteration, a good initial approximation may be needed to get fast convergence
for k = 1 (the coarsest grid). This has been achieved by using a continuation
process where we solve first for a small norm II (j) II , then gradually
increasing it until the prescribed norm is obtained. Each time the norm is
increased, the solution of the previous step is used as an initial

300

approximation. In order to get a good initial approximation for the smallest-
norm problem, we have solved for the bifurcation point from the trivial branch
of solutions.

3.5 Solving for the Bifurcation Point

ii is

At a bifurcation point (<f) , V^), the linearized problem of (3.1) must
have a zero eigenvalue, and the corresponding eigenf unction gives rise to a
second branch of solutions. Since (fi = is a solution for any Vq, we may
try to find a bifurcating branch from the trivial one (0,V„). The
linearized equations around (0,V„) are given by

W + rf- W ) + 4V^ a^(0)W =0, in f2 (3.8a)

xx'-rr-'rO

W = 0, on an. (3.8b)

If there exists a bifurcating branch from the trivial one (0,Vq), equation
(3.8) has a solution (W ,V-^) with IIW 11- = 1 where II Ij- denotes the L2
norm.

We discretize (3.8) in a way similar to the discretization of (3.1). The
constraint

IIW^II^ = 1,

is added to ensure a non-zero solution to the problem. The process of solving
the eigenvalue problem is identical to the process of solving (3.2), (3.4).

301

Once this linear eigenvalue problem is solved, we can use ^ = ±eW as

an initial approximation for our original problem with a prescribed norm of

e. The sign is chosen such that ^f. has negative values, to ensure that the

2 "
total streamf unction i() = _ + <(, will have closed streamlines with negative

values (the bubble).

4. NUMERICAL RESULTS

Experiments were performed with equations (3.2), (3.3) using FMG
algorithm of Section 3.4.2. In these experiments the domain was

n = {(nh, Zh), < nh < 5, < £h < 2}.

Three levels were used in the multigrid algorithm where the finest grid

problem has mesh size 1/16. On the coarsest level 20 relaxations were

performed while on finer grids v^^ = V2 = 3, y(k) = 4. In all numerical

experiments 1^ = I^~ is injection, I^~^ is bilinear Interpolation, and

n, . is bicubic interpolation.

Tables I-IX contain the Lo-norm of the residuals and the values of v3

^

at the end of each cycle on the finest grid. Cycle #0 refers to the
approximation obtained from the previous level as an initial guess. Figures

1-9 show the streamlines (contours of ^) for the different cases. The value

* *

of Vq, the swirl parameter value for which bifurcation occurs is V„ = 1.0069

(computed on coarsest level).

The experiments clearly show that the multigrid method suggested is very

efficient. In fact, as seen by the convergence history for V^ , it is enough

302

to take Y(k) = 2, instead of y(k) = 4, I.e., by 2 FMG cycles the problem is
already solved.

The results show that bigger bubbles are obtained for smaller swirl
parameters, contradicting to what one would expect. This may be the result of
the assumption made in the model, that the same functional dependence of
k, (JL) on ij) holds inside as well as outside the bubble. A future study will
investigate this point by solving the full systems (2.1), making no extra
assumptions.

303

REFERENCES

[1] J. H. Bolstad, H. B. Keller, "A Multigrid Continuation Method for
Elliptic Problems with Turning Points," to appear in SIAM J. Sci. Stat.
Comput.

[2] A. Brandt, Multigrid Techniques; 1984 Guide with Applications to Fluid
Dynamics . Monograph available as GMD-Studie No. 85, GMD-FIT, Fostfach
1240, D-5205, St. Augustin 1, West Germany.

[3] M. M. Hafez and M. D. Salas: "Vortex Breakdown Simulation Based on a
Nonlinear Inviscid Model," Proceedings of ICASE/NASA Workshop on Vortex
Dominated Flows , (M. Y. Hussaini and M. D. Salas, eds.), Springer-
Verlag, 1986.

[4] S. Leibovich: "Vortex Stability and Breakdown: Survey and Extension,"
AIAA J ., Vol. 22, No. 9, 1984, pp. 1192-1206.

[5] K. Stiiben and U. Trottenberg: "Multigrid Methods: Fundamental
Algorithms, Model Problem Analysis and Applications," in Multigrid
Methods , Lecture Notes in Mathematics, No. 960, (W. Hackbusch and
U. Trottenberg, eds.), Springer-Verlag, 1982.

304

Table I.

>^ll^ = .005

Table 11.

I(()^ll^ = .05

cycle # II Residuals II.

cycle #

I Residuals I

.362 (-1)

.95088

1

.986 (-3)

.96069

2

.843 (-4)

.96039

3

.148 (-4)

.96041

4

.745 (-5)

.96042

.948 (-1)

.68322

1

.232 (-2)

.68962

2

.251 (-3)

.68939

3

.113 (-3)

.68941

4

.918 (-4)

.68941

Table

III.

llc}."!!^ = ,

.11

cycle

#

II Residuals!

'2

v2
^0

.122

.59214

1

.233 (-2)

.54739

2

.215 (-3)

.54732

3

.615 (-4)

.54733

4

.542 (-4)

.54733

Table IV. \\^^\\^ = .15

cycle #

II Residuals II 2

'0^

.135

.48347

1

.243 (-2)

.48803

2

.168 (-3)

.48798

3

.474 (-4)

.48798

4

.425 (-4)

.48798

Table

V.

ii(j)"ir = .2

Table VI.

\\i>^\\^ = .4

cycle

#

II Residuals II2

^0^

cycle #

II Residuals II 2

v^

.150

.42902

• .192

.30435

1

.242 (-2)

.43301

1

.271 (-2)

.30725

2

.193 (-3)

.43294

2

.239 (-3)

.30719

3

.366 (-4)

.43294

3

.177 (-3)

.30719

4

.266 (-4)

.43294

4

.176 (-3)

.30719

305

Table

VII.

H^w^ = .6

Table

VIII.

, H^i|2 =

1.0

cycle

#

II Residuals II 2

V?

cycle

//

11 Residuals II

2

V?

.230

.24006

.295

.17139

1

.303 (-2)

.24335

1

.385 (-2)

.17303

2

.218 (-3)

.24231

2

.363 (-3)

.17302

3

.188 (-3)

.24231

3

.294 (-3)

.17302

4

.175 (-3)

.24231

4

.278 (-3)

.17302

Table

IX.

114.^11^ = 2.0

cycle

#

II Residuals II 2

<

.428

.10176

1

.701 (-2)

.10276

2

.777 (-3)

.10275

3

.584 (-3)

.10275

4

.574 (-3)

.10275

306

STREAMLINES

Figure 1. ||<(, « = .005, Vq = .96042.

STREAMLINES

Figure 2. Ilcji^ll^ = .05, Vq = .68941.

307

STREAMLINES

Figure 3. i^'^h^ = .11, V^ = .54733,

STREAMLINES

Figure 4. ||(j>^||^ = .15, Vq = .48798.

308

STREAMLINES

Figure 5. ll,j)^||^ = .2, Vq = .43294,

STREAMLINES

Figure 6. ^'^ll^ = .4, V^ = .30719.

309

STREAMLINES

Figure 7. ^^h^ = .6, V^ = .2A231,

STREAMLINES

Figure 8. ||<j,^||^ = 1.0, vj = .17302,

310

STREAMLINES

Figure 9.

Vi 9

ll(t) II = 2.0,

Vq = .10275.

311

CONSTRUCTION OF
HIGHER ORDER ACCURATE VORTEX AND PARTICLE METHODS

R. A. Nicolaides
Carnegie-Mellon University

ABSTRACT

The standard point vortex method has recently been shown to be of high
order of accuracy for problems on the whole plane, when using a uniform
initial subdivision for assigning the vorticity to the points. If obstacles
are present in the flow, this high order deteriorates to first or second-
order. This paper introduces new vortex methods which are of arbitrary
accuracy (under regularity assumptions) regardless of the presence of bodies
and the uniformity of the initial subdivision.

This work was supported by the Air Force Office of Scientific Research under
Grant AFOSR-84-0137.

312

1. INTRODUCTION

There has been a growing interest recently in the theory and application
of point vortex methods to the numerical solution of the incompressible Euler
and Navier-Stokes equations. The impetus for the Euler case stems from the
basic work of Dushane [6], Hald and Del Prete [7], and Hald [8], the Fourier
analysis of Beale and Majda [1], [2], [3], and the Sobolev space approach of
Raviart [12] and Cottet [4]. A recent paper by Cottet and Gallic [5] extends
the latter approach to linear Burger's type equations with "viscosity"
accounted for by splitting the convection and viscous parts and using a
Green's function for the viscous computation. A method for introducing
viscosity into particle methods for compressible flows is given by Monaghan
and Gingold [9]. See also [10] and [11]. Apart from the first three of these
references, the authors all obtain high order of accuracy error estimates,
limited mainly by the regularity of the exact solution of the continuous
equations. Unfortunately, the possibility of obtaining this accuracy is
dependent on the existence of expansions similar in nature to the Euler-
MacLaurin sum formula. If, for any reason, it is not possible to assert the
existence of such expansions, the accuracy drops to first- or second-order,
depending on the exact details of the algorithm and which errors are being
estimated. If general boundaries (bodies) are present in the flow field, or
if the initial subdivision of the flow field is not uniform, the necessary
expansions will most likely cease to exist. Then questions arise as to how
higher-order schemes may be constructed, and more important whether it is
worthwhile to use them in view of the extra expense which is involved. The
purpose of the paper is to give some possible answers to these questions.

313

In Section 2, the basic equations are given, and the simplest particle
method is defined for comparison with some higher-order schemes. These
schemes are introduced in Section 3. There, three methods for generating
schemes of arbitrary accuracy are provided. An appendix contains some
technical results about solving scalar hyperbolic equations with
distributional data.

This paper is of an algorithmic nature and does not contain numerical
results or precise error estimates. These will appear elsewhere.

2. MODEL PROBLEM

The incompressible Euler Equations in vorticity-velocity form are

u^ + (uco)^ + (voj) = "j (2.1)

\ in 1?
div(u,v) = : curl(u,v) = to J (2.2)

with initial condition

tj(x,y,0) = a)Q(x,y). (2.3)

The basic ideas for constructing higher-order schemes will be shown for (2.1)
and (2.3), with (u,v) assumed given. For these linear problems it is not
necessary to assume that (u,v) is solenoidal.

In this setting, we will now define the basic particle (or point vortex)
method. Subdivide the plane into squares of side h, number the squares
1, 2, 3,... in some convenient way and define a distributional approximation

314

to oIqCx.y) by

"Oh^'^'y) = I h a)(x^,y^) 6(x-x^, y-y^) (2.4)

where (x^,y^) denotes the center of the i*-*^ mesh square, and 6(x-x ,y-y )
denotes the Dirac delta function with pole at (xj^,y^). Now solve (2.1) and
(2.2) with ci)Q(x,y) ■<- a)Qj^(x,y). The well known solution to the latter
problem is the distribution

tOj^(x,y,t) = I h^ a)(x^,y^) 6(x- X(x^,y^;t), y - Y(x^,y^;t)) (2.5)

where X(xj|^,yj^,t) denotes the solution of the characteristic equation

dX/dt = u(X,Y,t) x(0) = x^

and correspondingly for Y.

No use is made of the uniformity of the mesh in deriving (2.5). For a
nonuniform mesh, h in (2.5) is the area of the appropriate mesh square. In
the error formulas below, h denotes the largest mesh length.

It is immediately clear from this definition that the particle
approximation is non-dissipative , in the sense that no artificial viscosity is
introduced because after the discretization of the initial condition is made
(2.1) is solved exactly. In practice some ODE solver must be used to compute
the trajectories, but in theory its error can be made arbitrarily small. This
principle, of solving the exact equation with approxlmte data, seems to be
common to particle methods generally and distinguishes them from finite

315

difference and finite element methods. The latter, at least, solves an
approximate equation with exact data.

A rigorous error analysis of the method just defined can be found in
[12]. This analysis is too complicated to reproduce here. Nevertheless, we"
need some simple guide to compare the accuracy of various schemes. It seems
reasonable to look at the difference u- - oj^, against a test function as a
measure of "truncation error" since it is the only error made. Thus we
define, for a given method of approximation and a given function tOp, with
compact support fi (where area (n) = 1 say)

'^hW = // (wq - WQj^)(t'dxdy. (2.6)

Here, the integration is performed over TSr . The restriction that a)_ has
compact support is a matter of convenience rather than necessity and could be
replaced by sufficiently rapid decay at large distances from the origin.
As an example, consider (2.4). Then we find

T^W = // Wq (t-dxdy - I h^(a)Q *)(x^,y^). (2.7)

This shows that a midpoint rule numerical integration is being used to
approximate the integral, and under smoothness conditions it follows that as
h -»■

Tj^(<f.) = O(h^).

Clearly, higher-order integration formulas can be compared with each other on

this basis. For a 2 x 2 product Gauss rule in each element, for example, we

have T, = 0(h ).
n

316

Next, recall the important fact that In the nonlinear case it is
necessary to compute the velocity field at each timestep by solving (2.2).
Assume that this is to be done using the Green's function. Let W denote the
number of arithmetical operations required to compute the velocity field at

9

each particle position. If there are N particles, then W == CN /2, for
some constant C. Below, we will use W as a standard unit of work to
compare various new algorithms. For the Gauss case therefore we have a work
count of 16W. From this we see that use of a higher-order rule does not
necessarily assure a greater computational efficiency for typical values of
h. In the next section, methods for obtaining high-order accuracy without
such a large increase in the cost of the computation are defined.

3. HIGHER ORDER METHODS

The preceding remarks suggest that increasing the order of accuracy by
adding more integration nodes may not be a good idea. It is natural to try to
do the same thing by increasing the amount of information associated with each
node. Specifically, in this section we shall associate with (x.,y.),
order distributions of the form

mth

M^(x,y) = I w^^ d" 6(x-x^, y- y^). (3.1)

|a|<m

In (3.1), which generalizes the simple 6 functions in (2.4), a denotes a
multi-index, and (xj^.y^) e I?. Choice of the weights w. and the nodes
(Xj^,y£) can be made in many ways. We shall give three methods in this
section.

317

Method 1 (Direct Integration) ;

In this method, (x^,y^) are the corners of the elements, each of which
has associated with It an expansion of the form (3.1). The weights In the
expansion are chosen so that when a}„. Is substituted into (2.6), the second
term gives a rule for Integration of the function (a3„ (}>), Involving its
values along with those of its derivatives through order m at the nodes. We
shall consider the cases m = and m = 1 In more detail.

Let m = 0. A rule for a square of side h with corners at P, Q, R, S
which is exact for bilinear functions is

// f dxdy = (h^/A) (f(P) + f(Q) + f(R) + f(S)). (3.2)

2
Using this as a composite rule Implies the choice w = h w(x ,y ) sc

that we define

M^(x,y) = h^u(x^,y^) 6(x-x^, y-y^). (3.3)

Since this gives a rule which is locally exact for linear functions but not
for all quadratics its accuracy is 0(h2) in the sense of (2.6) while the
work is IW. This is essentially no different from the mid-point rule. In
fact this rule is clearly analogous to the trapezoidal rule.

For a quadrilateral mesh, a bilinear mapping can be used to map the
quadrilaterals onto a standard square in which (3.1) can be used. In some
circumstances it may be desirable to use a triangular mesh Instead of the
quadrilateral one. An 0(h2) rule for triangles analogous to (3.1) can then
be used, avoiding the need to map the domains.

318

Now let m = 1. Analogous to (3.2) we have the formula

// f dxdy = A(f(P) + f(Q) + f(R) + f(S))

+ B(-yp) + f^(Q) + yR) - ys)) (3.4)

+ C(-fy(P) - fy(Q) + fy(R) + fy(S))

where A = h /4, B = C = h-^/24, and P, Q, R, S denote the corners of the
square -h/2 <. x, y <^ h/2 labelled counterclockwise starting from the top
right. Analogous to (3.3) there is the expression

M^(x,y) = I w^^ d" 6(x-x^, y-y^). (3.5)

kill

In (3.5), the coefficients are computed from the composite rule based on
(3.4). For the uniform square mesh we are using for illustration, the weights
are

"iOO = A'tOQ(x^,y^) + B'a)Q^(x^,y^) + C'o) (x^,y^)

^ilO = -B''OQ(x^,y^)

''iOl = -C''OQ(x^,y^).

(3.4) is exact for cubic polynomials. It follows that this method is accurate

,4

in the sense of (2.6) to O(h^). To compute work units for this scheme.

we

observe that although there are only = N particles there is some extra work
associated with computation of derivatives of the velocity kernel. It turns

319

out that for this scheme the work units are < 2 y W, a satisfactory figure.
There is also some additional work required for computing the coefficients of
the derivatives in (3.1). This amounts to having to integrate two more
systems each of two odes, in addition to the characteristic odes (see
appendix) .

As in the previous case, rather than use a quadrilateral mesh it might
sometimes be better to use a triangular one.

For a square mesh, the m = 1 scheme just discussed has an interesting
property in the uniform case. This is the following: due to cancellations,
the composite rule has weights of zero attached to the derivative unknowns at
interior vertices. Hence the higher accuracy is achieved by corrections at
the boundary. But this implies the use of a Euler-Maclaurin type expansion.
Thus, if (ill}) has s continuous derivatives in W and compact support, by
using nodal derivatives up to this order we can get accuracy 0(h^"*'^) merely
by using the m = scheme, since this is what the composite scheme reduces
to on a uniform mesh in that case. This is another way to look at the results
of [1] - [3].

Method 2 (Finite Element Approach) ;

The approach here uses a nodal finite element basis in the following
way: let {\li^ } |a| _< m, i = 1,2,«««, be the standard nodal basis functions
associated with the i node (xj^,y^) of a triangulation of the plane with
maximum edge length h. These functions satisfy conditions of the form

D^ ^, (x.,y.) = a"?,
^ia^ j'-'j ij'

320

a8
where A,, is a Kronecker delta. Then we define w. as
ij ia

w, = (-l)'"'!// i}». (x,y) a)„(x,y)dxdy (3.6)

io ^ ' •'J ^ia

where the integration is over the whole plane. We now have

// a)Qj^(x,y) p(x,y)dxdy = jj I I w^,^ d"6(x-x^, y-y^)

^ l«|<tn

X (|)(x,y)dxdy, V ({> € C™ (IE?)

(3.7)

= I I (-1)'°'' w^^ d" <t.(x^,y^)
^ l«|<m

= // a)Q(x,y) <)) (x,y)dxdy

where <j) is the finite element interpolant of (}> on the given
triangulation. Equation (2.6) then becomes

Tj^((t)) = // a)Q(((. - <t)^)dxdy. (3.8)

Since the error |<t> - <l> | is formally 0(h'^ ) where r is the degree of
the highest order full polynomial space used, we can say here that t, is of
this order.

This type of scheme differs from direct integration schemes in that no
approximation of oj^ is made. The test function only (often a convolution
kernel in practice) is approximated and the result is integrated exactly.
Because of this property, the rigorous error estimates for these methods

321

require minimal regularity on m unlike the direct integration methods where
to achieve high accuracy requires a)_ to have several smooth derivatives
throughout I?. The 0(h^+l) estimate is in fact valid even if we know
only (^0^^ (^)« If Wq has extra regularity it can be exploited to get
higher accuracy by going to negative norm estimates of the finite element
error. Smoothness of <)) , however, is certainly required.

Two examples analogous to those considered above are the case of
continuous linear elements on triangles, for which we can expect O(h^)
accuracy with IW work units, and full cubics - defined in terms of
derivative unknowns at vertices, and function values at vertices and centroid
for which the work will be somewhat larger than the values used so far (about
10 Y W units).

In general, the full range of finite element spaces is available for use.

Method 3 (Taylor /Moment Expansions) :

Here we begin by subdividing the plane into arbitrary elements with mid-
side nodes and arbitrary element shapes allowed in principle. Next, we define

"l' "2' ''ia ° (-1> ' //(x-^i) (y- y^) a)Q(x,y)dxdy (3.9)

in which (xj^,yj.) is an arbitrary point within the i*"^ element, and the
integration is over the ith element. The w are proportional to the
moments of w^ restricted to the i*"" element, about (xj^,y^). It follows as
above, that

// a)Q^^(x,y) (t.(x,y)dxdy = // a)Q(x,y) .t.^""^ (x,y)dxdy (3.10)

322

where ^ (x,y) is the piecewise polynomial function, in general

t* v»
discontinuous, equal in the i element to the Taylor expansion of ({)(x,y)

through m*" order terms, about the point (x^jy^^). In this sense the local

moment expansion defined by (3.1) and (3.9) "dualizes" into the local Taylor

expansion about (x^.y^).

To get the accuracy of this scheme, we substitute into (2.6) to find that

T^(<t.) = // a)Q(<i. - (frf^bdxdy

so that denoting by h the largest linear dimension of the elements, we
obtain accuracy 0(h™'^^).

The moments method also needs only minimal regularity on a)„ for full
accuracy to be obtained. In practice, if m = 1 the point (xj,yj) should
be chosen to be the center mass of oi^ because then w, =0 for |a| = 1,
so we get second-order accuracy for the same work as with the lowest-order
scheme. Using quadrilaterals for elements, with N vertices there are
approximately N elements and so N particles. For 0(h) accuracy the
interaction work count is 5W, and for 0(h) is 8W.

4. FURTHER REMARKS

There should be no difficulty in extending the ideas of Section 3 to
three-dimensional particle methods of the kind suggested in [1] - [3] and
[12].

Rigorous analysis using the Sobolev space setting has been carried out
for both the finite element and moment expansion methods.

323

So far an insufficient amount of computation has been done to verify the
error estimates and decide about the efficiency of the various methods.

ACKNOWLEDGEMENTS

Thanks to Chichia Chiu and Shenaz Choudhury for their help with this
paper.

324

APPENDIX

A framework for finding distributional solutions of (2,1) with initial
condition (o-, = d" 6(x-Xp^, y~yr)) !«! £ ™ can be obtained starting from the
following considerations. Let X(xQ,yQ;t) and Y(xQ,yQ;t) denote the
characteristic curves of the equation (2.1); here, t parameterizes the curve
and the generic point (xqjYq) denotes its origin at time t = 0. X and Y
are computed by solving the ordinary differential equations

dX/dt = u(X,Y,t) dY/dt = v(X,Y,t)

X(0) = Xq Y(0) = yQ.

At time t, let J(xq, yg; t) denote the Jacobian of the flow map
$ : (xQ,yQ) •♦■ (X,Y). The (nonlinear) case of most interest from the fluids
viewpoint has u^^ + v = 0, in which case J(xQ,yQ;t) =1. We can obtain a
formal analytical solution to (2.1) and (2.3) by writing the equation in terms
of the material derivative as doi/dt = 0, integrating this equation over an
arbitrary domain moving with the velocity field (u,v), say n(t), and then
using the transport theorem to write

d/dt // a)(X,Y,t)dXdY = 0,

n(t)

from which it follows immediately that

// u)(X,Y,t)dXdY = // a)^(x,y)dxdy.

n(t) n(o)

325

Changing the variables on the right-hand side to X and Y respectively and
recalling the arbitrariness of n(t) now gives

a)(X,Y;t) = a3Q(x(X,Y,t), y(X,Y,t))j"^ (X,Y;t) (A.l)

where (x(X,Y,t), y(X,Y,t)) is by inverting the equations X = X(x,y:t), Y =
Y(x,y;t). The existence of a unique solution to these equations follows from
ode theory provided u and v are smooth. Reversing the steps, it follows
that (A.l) satisfies (2.1) given the required regularity of u, v, and a)_.

Let (|) e C™(I^); multiplying (A.l) by <t), integrating and changing the
variables on the right to x and y we have

// a)(X,Y,t) <i,(X,Y)dXdY = // a)Q(x,y) <l,(x(x,y; t), Y(x,y,t))dxdy, (A.2)

or alternatively

<u, (})> = <toQ, <f.o(X,Y)> (A. 3)

where o denotes composition. If X(«,», t) and Y(»,«, t),
Y(.,., t)ew'"'^^'"(]E2) (or € C^'^^d^)), ¥ < t < T, then the right-side of
(A. 3) makes sense even if w^ -<- u^^ = d" 6(x-Xq, y-yQ)|a| < m. Thus a

distribution oj is defined on (p^^^d?) by (A. 3). Therefore, we can pose
the problem of finding o), satisfying

<a)^, (j)> = <i^^^, <j)o(X,Y)> V ({. € C^""^ (]^j. (A. 4)

326

A solution w, to (A. 4) is given by

0),

(X,Y) = D° 6(X - X(x,y;t), Y - Y(x,y;t))

(A. 5)
x=Xq. y=yQ

the purely formal differentiations being performed w.r.t. x and y. Proof
that (A. 5) satisfies (A. 4) is by direct computation.

If |a| = we recover the solution given in Section 2. Consider the
case with |a| = 1. Equation (A. 5) gives

(A.6)

using the abbreviation Xq for X(xQ,yQ;t) and similarly Yq. If the
initial condition is

''ho = ^10 ^x(^-^o' y-^o^ ■" ^01 ^(^^-^0' y-^o)'

then the solution to (A. 4) of the required form as given by (A.6) is

'^h = ^0^^^ ^X^^-^0' Y-^o) ^ %l^^^ ^Y^^-^O' ^- ^o)
where

(A. 7)

^01^'^ = ^0 ^K'^O'^) •" ^01 V^O'^O'^^

327

Letting M denote the matrix

X X
X y

Y Y
X y

differentiation of the characteristic equations shows that

dM/dt = V(u,v)M

and the initial condition for this system is M(0) = 1, the identity matrix.
It will be necessary to solve this and analogous systems for the higher-order
cases in order to compute the numerical approximations. Having solved it,
aj^gCt) and aQj^(t) are given by (A. 7).

328

REFERENCES

[1] J. T. Beale and A. J. Majda, "Vortex Methods 1: Convergence in Three
Dimensions," Math. Comp. , Vol. 39, 1982, pp. 1-27.

[2] J. T. Beale and A. J. Majda, "Vortex Methods 2: Higher Order Accuracy
in Two and Three Dimensions," Math. Comp. , Vol. 39, 1982, pp. 29-52.

[3] J. T. Beale and A. J, Majda, "Higher Order Accurate Vortex Methods with
Explicit Velocity Kernels," J. Comp. Phys. , Vol. 58, 1985, pp. 188-208.

[4] G. H. Cottet, "Methodes Particulaires Pour L'equation D'Euler dans Le
Plan," These de 3e cycle, Univ. P. et M. Curie, Paris, 1982.

[5] G. H. Cottet and S. Gallic, "A Particle Method to Solve Transport-
diffusion Equations," Report 115, Centre de Math. Appl., Ecole
Polytechnique, 1985.

[6] T. E. Dushane, "Convergence of a Vortex Method for Solving Euler's
Equation," Math. Comp. , Vol. 27, 1973, pp. 719-728.

[7] 0. Hald and V. M. Del Prete, "Convergence of Vortex Methods for Solving
Euler's Equations," Math. Comp. , Vol. 32, 1978, pp. 791-809.

[8] 0. Hald, "Convergence of Vortex Methods II," SIAM J. Numer. Anal ., Vol.
16, 1979, pp. 726-755.

329

[9] J. J. Monaghan and R. A. Gingold, "Shock Simulation by the Particle
Method SPH," J. Comp. Phys ., Vol. 52, No. 2, November 1983, pp. 374-389.

[10] J. J. Monaghan and R. A. Gingold, "Kernel Estimates as a Basis for
General Particle Methods in Hydrodynamics," J. Comp. Phys ., Vol. 46, No.
3, June 1982, pp. 429-453.

[11] J. J. Monaghan, "Why Particle Methods Work," SIAM J. Sci. Stat. Comput .,
Vol. 3, No. 4, December 1982, pp. 422-433.

[12] P. A. Raviart, "An Analysis of Particle Methods," CIME course. Numerical
Methods in Fluid Dynamics, Como (1983).

330

PSEUDO-TIME ALGORITHMS FOR THE NAVIER-STOKES EQUATIONS

R. C. Swanson
NASA Langley Research Center

E. Turkel

Tel-Aviv University, Israel

and

Institute for Computer Applications in Science and Engineering

ABSTRACT

A pseudo-time method is introduced to integrate the compressible Navier-
Stokes equations to a steady state. This method is a generalization of a
method used by Crocco and also by Allen and Cheng. We show that for a simple
heat equation that this is just a renormalization of the time. For a
convection-diffusion equation the renormalization is dependent only on the
viscous terms. We implement the method for the Navier-Stokes equations using
a Runge-Kutta type algorithm. This enables the time step to be chosen based
on the inviscid model only. We also discuss the use of residual smoothing
when viscous terms are present.

Research was supported in part by the National Aeronautics and Space
Administration under NASA Contract Nos. NASl-17070 and NASl-18107 while the
second author was in residence at ICASE, NASA Langley Research Center,
Hampton, VA 23665-5225.

331

I. INTRODUCTION

The solution of the compressible Navier-Stokes equations for flow about
two- and three-dimensional complex aerodynamic configurations is still a time
consuming problem on today's supercomputers. The resolution of the boundary
layers requires the use of very fine meshes in the neighborhood of solid
bodies. For a typical viscous flow the mesh can be several orders of
magnitude finer (depending on the Reynolds number) than that required for an
inviscid calculation. As an example, using a C-type mesh about an NACA 0012

airfoil, a typical mesh spacing near the body in the normal direction for an

_2
inviscid calculation is 1 x 10 chords. For a laminar viscous calculation

3 -4

with Re = 5 X 10 , this minimum cell height would be about 6 x 10

chords. For a turbulent calculation using an algebraic turbulence model and

with Re = 3 X 10 , the minimum cell height would be about 8 x 10

chords. In all cases a typical chordwise spacing at the midsection of the

-2
airfoil is about 5 x 10 chords.

Using an explicit method this fine mesh reduces the time step, due to

stability requirements, that can be used. The time step restriction is caused

by two factors. One contribution is due to the effect of the finer mesh on

the inviscid portion of the calculation. When using an explicit method this

reduction of the time step cannot be avoided without using a coarser mesh. It

follows strictly from the need to include the entire domain of dependency in

the numerical algorithm. Use of a local time step allows faster convergence

to a steady state, but it does not remove the requirement to satisfy the

convection stability condition in a local sense. A second difficulty is

caused by the viscous terms. For an explicit method the time step is now

dependent on the square of the mesh size rather than just the mesh size as

332

occurs for inviscid flow. Thus, even for a high Reynolds number flow the
viscous time step will dominate when the mesh is sufficiently fine. In all
these cases the use of an implicit scheme will alleviate the difficulties. In
some ADI methods the Jacobian of the viscous terms is not used in the implicit
portion of the code in order to improve the speed of the calculation [7]. We
thus conclude that for both explicit and many implicit codes it is
advantageous to account for the dependence of the time step on the viscous
terms.

In this study we shall only discuss steady state problems which are solved
by a pseudo time-dependent method. Hence, we can change all time derivatives
as long as the steady state solution is not affected. One common device is to
use a different time step in each zone. It is easier to calculate this local
time step based on the inviscid equations. This provides an additional reason
to eliminate the dependence of the time step on the viscous terms.

In this study we shall analyze a method used by Crocco [4] and also by
Allen and Cheng [2]. They claim that the new scheme is unconditionally stable
for a simple diffusion equation. We will show that in effect the scheme is a
standard Euler forward-in-time central-in-space scheme. The time is
artificially slowed down so as to satisfy the stability criterion. We then
extend this scheme to the compressible Navier-Stokes equations using a Runge-
Kutta scheme [9], This modification enables us to choose our time step based
on the inviscid equations. The modification automatically reduces the local
time step in regions where the viscous time step is of importance. This
enables us to use the inviscid time step in the far field while automatically
accounting for viscous effects in the boundary layer. We will also look at
residual smoothing for the heat equation.

333

II. SCALAR EQUATION

In this section we analyze and extend a scheme for the Navier-Stokes
equations proposed by Crocco [4] and Allen-Cheng [2]. This scheme was also
analyzed by Peyret and Viviand [6] and Roache [8], and we will extend their
analysis.

We first consider the heat equation

w = ew . (1)

t XX '

The forward time centered space or Euler approximation to this scheme is given

by

n+1 n . eAt r n _n,ni ,_.

This scheme is stable if

V = -^ < 1/2 .or At < -^ (3)

Crocco, and Allen/Cheng introduce the inconsistent scheme

w^-^1 = w!^ 4- -^ (w'?^, - 2w"^l + w'? J.

(4)

This scheme is unconditionally stable. If we are only interested in the
steady state, then (4) yields the correct steady-state solution. We now
rewrite (4) as

n+1 n
w *■ w
_J j^ _ e^ (-n _ ry ^ ^ ^ ^ _ 2e ^ n+1 _ n^

At

(Ax)

(-n „n,n-> 2e^ n+1 n-v
t" [w. , , - 2w. + w. ,1 - 75- ( w. - w.

334

or

/- 1 , 1 ■> c n+1 n-v Ez-n „n.n^ ...

(^-»-^)(wj -w.)=-— ^(w.^^-2w. +Wj_^) (5)

with

.,.^. (6)

Thus, for this model problem the Crocco scheme is Identical with the Euler
scheme (2) with an artificial time step At given by

At At ,, ^2 • ^'-'

e (Ax)

Thus, the unconditional stability is achieved by slowing down the time

2
process. Note that as At ■»• «, At ->■ (Ax) /2e, i.e., the stability limit for

the Euler method. So choosing a large time step for (4) is equivalent to

choosing At at the stability limit for (2), and we have merely scaled the

time. This can also be derived from the modified equation given in [6]. If

e or Ax is not constant, this also introduces a local time step.

We next consider the convection-diffusion equation

w^ = aw + ew . (8)

t X XX

The Crocco scheme now becomes

n+1
w. - w
J

n (• n n 1
a( w. , , - w. ,

1 = —Jll izll + _E_ fw" - 2w""'^ + w" 1

(Ax)

(9)

or

Q - irK-r' - "P - °'"°^^"J^''^ ^ ^ (Vi - 2"J * vi) <io)

335

with At given by (6). Thus, again this is equivalent to the Euler scheme
with a time scaling that depends only on the viscous terms. Allen and Cheng
utilized this scheme within a time-marching scheme proposed by Brailovskaya
[3], We generalize this by considering a general N-stage Runge-Kutta scheme.
Consider the two-dimensional equation

w^=Hw+e,w+e. w (11)

t 1 XX 2 yy

where Hw describes the hyperbolic or first-order terms. In [9] we describe
a Runge-Kutta scheme where the viscous terms are frozen for all the stages.
This is similar in philosophy to the Brailovskaya scheme. Using the Crocco
formulation the (K + l)-st stage becomes

(K+1) n
w. , - w. ,

«K+1 ^^

H w^^^ + -^ fw" - 2w^^-*-^) + w" 1

(12)

G,

*^K.m-^",T'*"".^-i5- •^-0.^" •.»-'■

This reduces to a Runge-Kutta scheme

where Hq, Pq are the approximations to the hyperbolic and parabolic parts
respectively and

1 1 2^ 2^2
""S ''^ (Ax)^ (Ay)^

336

We slightly generalize (14) by redefining At by

AT- = aF ^ 2K(-l4 + -^) (15)

^•"e ^^ (Ax)^ (Ay)^

where k is a constant that we can choose. The form of (15) no longer
follows directly from the Crocco formulation. Instead k will be chosen
based on a stability analysis.

We choose At in (12) or (15) based on the hyperbolic (inviscid)
stability condition. We then find At from (15) and advance to stage
(K + 1) using the Runge-Kutta scheme (13).

The constant k in (15) can be chosen so that we recover the parabolic
stability limitation when Hp = 0. The exact value of k depends on the
coefficients ctj. in the Runge-Kutta formula. In order to see this more
clearly we revert to the one-dimensional convection-diffusion equation (8).
We replace all space derivatives by second-order central differences while the
time derivative is kept continuous. We therefore have

al w, , , - w. , J
w J-^^ JZll + __L_r„" _ 2w" + w" 1. (16)

We Fourier transform (16) to get

w^ = Xw (17)

with

HO = - -^^ (1 - cos C) + i^ sin ? < ? < 277. (18)
(Ax)"^ ^'^ ~ ~

A Runge-Kutta scheme for (16) or (17) is stable whenever z(5) = X(5)At lies
within the stability domain that depends on a, ,«««,a„ for all < C < 2t7.

337

We consider the stability domain for the four-step scheme with a, = 1/4,

a^ = 1/3, a_ = 1/2, a, = 1. This scheme has a stability condition along the

imaginary axis of max|z| <_ 2/2", i.e., for a hyperbolic problem (e = 0)
aAt ^

Ax

<^ 2/2. Along the negative real axis the stability condition is

2eAt

|z| < 2.8 and for a parabolic problem (a = 0) 5- < 2.8. Hence for this

(Ax)^ '^

case we would choose k in (15) as k = 1.4. We define the cell Reynolds
number as

\ - "f- (»)

The previous analysis shows that the Runge-Kutta scheme is stable for R^ =
and R, = ». We do not have any proof that the scheme is stable for all R^.

III. NA.VIER-STOKES EQUATIONS

We now discuss the implementation of these ideas to the two-dimensional,
compressible, Navier-Stokes equations. The extension to three dimensions is
straightforward. We first consider the conservation form in Cartesian
coordinates. We express the equations in the following form

P, -Hj

9x ■' 9y

(p,)^ . H3 + „ i!i t (X . „) |!|^ . (X + 2„) 1^

9x •' 9y

(20)

338

9x 8y

2 2

_l_/•\J,o^ 9u, 9v

+ (X + 2vi)u — ^ + pv — r-

9x 9x

where

.(X.,)[v|!h_..|!|_]

+ yu ^ + (A + 2y)v -^
9y^ 9y^

« - 17 _ (pu) + (pv)

2p ♦

and Hj denote first derivative terms (including the artificial viscosity and
also the viscous dissipation function). The coefficients of viscosity (y
and X), Y the specific heat ratio, and the Prandtl number Pr are all
assumed (for the analysis) to be locally constant.

In deriving our results we shall ignore all cross derivatives (see, e.g.,
[1], [2]). Based on our previous analysis we add the following terms to the
standard Runge-Kutta scheme.

Ap = Kj

Mpu) = K, - 2[hJL^ + _L_^] A(^ «At
(Ax)^ (Ay)^ P

A(pv) = K, - 2[-^ + iL±4] A(PV)_ ^^^
(Ax)^ (Ay)^ P

(21)

A(pE) = K, - 2^ [- ^ (-i-^ -. _L^) + ilJli^ + -i4]A(pu)aAt
P ^'^'' (Ax)^ (Ay)^ (Ax)^ (Ay)^

339

where Aw = w - w and K denote the usual space derivative terms.
For simplicity we have chosen k = 1, and a denotes the constant in the
Runge-Kutta scheme (28). Thus the density equation is unchanged. The second
and third equations can be solved directly for A(pu), A(pv). Once A(pu),
A(pv) are known the last equation can be solved for A(pE). As before these
corrections imply an effective time step which automatically accounts for the
viscous time step. In this case the effective time step differs for each
equation.

We finally consider the Navier-Stokes equation in body fitted
coordinates. This can be done either in a finite volume scheme or by using
transformations. The result is the same in either case [9], and so we shall
use a transformation for ease of presentation. Let 5 = 5(x,y), r\ = ri(x,y)
be the body fitted coordinates. We choose the coordinate scaling so that
A? = An = 1. The Navier-Stokes equations (20) now become

Pt = «1

(pu)^ = H + [(X + 2m)cJ + ii?2] i-| + [(X + 2u)n^ + pn^) ^

8^v S^v

+ (X + li)5 5 — J + (X + )j)ri ri — ^ + crossterms

(pv)^ = H3 + [u^l + (X + 2^^)d] ^ + [un^ + (X + 2y)n^] ^

■^9? ^ 8n

340

2 2

+ (X + vK £ -^^ + (X + y)n n„ -^ + crossterms (22)

2 „ „ .2

(P«,-H4-f?[(«'*«y)B^(\*"y)Ml

2
+ [(X + 2y)u5j + (X + u)vE E + yu^J] ^

X y y 95

2

+ [(X + 2p)unJ + (X + y)vn n + pun^] -^

^ ^ ^ 9ti

2
+ [pv?^ + (X + yW 5^ + (X + 2p)vc2] -^
X X y y 95^

2

+ [pvTi + (X + p)ur) n„ + (X + 2p)vii ] — ^ + crossterms

X y y g^2

where H. are first derivative terms and we have ignored all second cross
derivative terms. As before this generates an appropriate correction term to
the Runge-Kutta scheme. Equation (21) Is now replaced by

Ap = Kj^

A(pu) = K, - 2[(X + 2p)4 + p5^ + (X + 2p)Ti2 + pn^l -^^^ aAt
•^ X y X y-' p

- 2(X + p)(c 5 + n n ) Aieil aAt
X y X y p

A(pv) = K3 - 2(X + y)(5^ ?y + n^ Tiy) ^^ (xAt

- 2[p?^ + (X + 2v)d + ml + (X + 2p)ti2] ALEII aAt
A y X V D

y p

(23)

341

A(pE) = K, - ^ (C^ + 5^ + n^ + Ti^)A(pE).aAt
4 pPr '■ X y X y

-2[-fI^(4 + 4..^np.<x.2„)uU^*nJ)

+ (X + y)v(C 5 + Ti n ) + uu(5^ + n^l] ^^^" aAt
X y X y *> y y-'-' p

„r V YP rp2 . p2 , 2 , 2<v , (--2 ^ 2-,

+ (X + p)u(5 5 + n Ti ) + (X + 2y)vr5^ + n^)] ^^^ -aAt
^x y X y ^ y y^-" p

where K. represents the standard finite difference terms.

As before the density equation is unchanged by the viscous correction.
Now, however, the two momentum equations are coupled together, unless the
coordinate system is orthogonal. As we have two equations for A(pu) and
A(pv), and we can easily solve these. To simplify the notation we define

z^ = 1 + 2aAt ^(^^2u)U' + nJ) + y(c^ + np]

,2 = 2a^(x + u)(5^5y + n^ny) (24)

z = 1 + 2£iAt ^^2 ^ 2^ ^ ^ ^ 2 ^ 2^^

4 p '■ ^ X . X-* >'".^y y-'J

and

D = (1 + Zj)(l + z^) - z

2
2*

Then

342

Ap = K^

A(pu) = -i— 1__1_1 (25)

A(pv) =

As before given A(pu) and A(pv) we can solve for A(pE) directly from the
energy equation in (23). We also note that if one uses the thin layer
approximation (dropping all second 5 derivatives and cross derivatives in
(22)) then these terms simplify slightly. In this case Ap, Apu, Apv are
still given by (25) with

, . 2aAt r,, „ V 2 2i
Zj = 1 +— ^ [(X + 2y)Ti^ + uTIy]

2aAt ,. ^ .
^2 =-^(^ + y)Tl^ Tiy

=^4 = 1+^ [V'\+ (^ + 2u)nJ] (26)

J = x_ y - X y_

and

^'^^^(^^^)w»«■K,

*.j.i.Ajr A Xy y "^ p

2[- J (g-)(nj + nj) + yvn2 + (x + y)un^ n + (x + 2u)un2] A^p::^. -aAt.

*• '^ '^ •'^ / X X y y-* p

343

IV. RESIDUAL SMOOTHING

As an alternative method of reducing the effect of the parabolic terms on
the stability of the scheme we consider residual smoothing. With this
technique one post-processes an explicit method with an implicit method. In
practice one post-processes each equation separately and each direction
separately so that only scalar tridiagonal matrices need be inverted. When
using a multistage Runge-Kutta method, one can apply the residual smoothing
after each stage, or at the end of the entire process, or any intermediate
permutation.

In [10] it is shown that one can construct such a scheme for a hyperbolic
equation so that the total method is unconditionally stable. It is further
shown in [10] that it is not efficient to use a very large At even ignoring
splitting errors. An optimal At is about two to three times larger than the
explicit time step. We now consider the process for a parabolic problem in
order to see the effect of viscous terms.

We, therefore, consider the heat equation

u^ = bu . (27)

t XX

We solve this equation by a k-stage Runge-Kutta scheme

u^^^ = u" + a AtQu

(£+1) n , .^„ (£) ^„QV

u ' = u + a , AtQu (28)

n+1 (k)
u = u

344

where a. .•••,a, are given coefficients with a, = 1. Q is a difference
approximation to u-,„. The amplification factor corresponding to (28) is

G = 1 + B^ AtQ + 32(At)^ q2 + ... + 3j^(At)'^ q'^ (29)

where ^i '^ ^ ^"^ ^o ~ ^ o-i %-o+i' ^ = 2,...,k. Q is the Fourier
transform of Q. Hence, for second-order central differencing

Q = - ^fa sin^(6/2) ^ (30)

uxr

Residual smoothing consists of updating a stage (A) by

(1 - aD2)Au^^^ = u^^^ - u" (31)

where D, is again a second-order central difference approximation to Uj^^^,
i.e., T>2 •*■ (1,-2,1). We now consider two possibilities. In the first we
apply (31) only after the final stage. Then the new amplification factor is

B, AtQ + B,(At)^ Q^ + ... + B, (At)*^ Q^

G(e) = l+-i 1 5 ^ . (32)

1 + 4a sin (e/2)

The second case we consider is applying (31) after every stage. The resultant
amplification factor is

02(6) = 1 + B^ AtR + 32^^*^)^ ^^ ■*■ ••* "^ ^^^'^^^ ^ ^^^^

with

R iL_^ . (34)

1 + 4a sin (e/2)

345

We now investigate the possibility that either of these schemes is
unconditionally stable. To investigate this we need only consider At
sufficiently large. We thus consider At ^ " with a ■»• ". Then (32) becomes

(-1) \[ 2 ^^^ ^J

G,(9) -^ 1 + i^^ . (35)

1 + 4ff sin (6/2)

We thus see that for k even, G (G) > 1 and so (28) - (31) cannot be stable
for At large. For k = 1 the scheme is identical with backward Euler for a
scalar one-dimensional equation and, hence, unconditionally stable. For the
second case we see that (33) has the same form as a standard Runge-Kutta
method with Q replaced by R, (34). Hence, it follows that the scheme is
stable whenever AtR is within the stability region of the scheme. As
At ->• ", so does a and so there is a cancellation between the numerator and
denominator; thus, AtR remains bounded as At increases. We thus conclude
that applying the residual smoothing after each stage can make the scheme
unconditionally stable even for a Runge-Kutta method with an even number of
stages.

We also see from the above argument that as At increases so must a. In
[9], [10] we show that for a hyperbolic equation

Uj. + aujj =

2
that a is proportional to (aAt/Ax) . For the parabolic problem (27) it

follows from (35) that a should be proportional to bAt/(Ax) . For the

combined convection-diffusion equation a will be related to the sum of two

such contributions.

346

It follows from (33), (34) that if we apply residual smoothing after every
stage then the stability polynomial has the same form as the original

A A

polynomial (29). The only difference is that Q is now replaced by R. From

A A A

(34) it follows that the ratio of Q to R is real. Hence, if Q is any

A

complex number then R lies along the same ray in the complex plane but with
a different amplitude. We therefore have shown that if the original scheme
was unstable for a given direction then residual smoothing cannot stabilize
the scheme. Furthermore, if the original scheme was conditionally stable then
by choosing a = a(At) sufficiently large we can make the scheme
unconditionally stable. We have thus shown

A

Theorem: Let Q be the amplification factor for any approximation to the
convection-diffusion equation and let (29) be the stability polynomial for a
k stage Runge-Kutta scheme. We now apply residual smoothing, (31), after
every stage of the scheme. If the original scheme was unconditionally
unstable then the new scheme is still unconditionally unstable. If the
original scheme was conditionally stable then the scheme with residual
smoothing can be made unconditionally stable by choosing a(At) sufficiently
large .

Hence, if the smoothing is applied at the end when solving a parabolic
equation, then the scheme can be unconditionally stable only when using a
multistage scheme with an odd number of stages. When the smoothing is done
after each stage, the scheme can be stabilized for a large. For a system
with a hyperbolic portion and a small parabolic contribution, e.g., high
Reynolds number Navier-Stokes , the residual smoothing is most effective with a

347

time step about twice that of the explicit convective portion. Hence, the
question of unconditional stability is somewhat academic. In practice [8] the
Runge-Kutta scheme for the Navier-Stokes equations is used with four stages
and with the residual smoothing applied after each stage.

V. RESULTS

In this section we present some results for viscous flow obtained using
the analysis of Sections II and III. We used a Runge-Kutta code to solve the
Navier-Stokes equations for two flows about an airfoil section. The details
of this code are discussed in [5], [9], [10], [11]. In these cases we
considered only the thin-layer form of the Navier-Stokes equations.

For the first case we computed laminar flow over an NACA 0012 airfoil with

a free-stream Mach number M of 0.5 and a Reynolds number Re of

3
5 X 10 . The angle of attack (a) of the airfoil was zero degrees. Half-
plane calculations were performed using a C-type grid consisting of 64 cells
in the streamwise direction and 64 cells in the normal-like direction. The
grid spacing at the airfoil surface was about 6 x 10~^ chords. The mesh
spacing in the streamwise direction over the central part of the airfoil was
AX = 0.05 chords. Results for this case are shown in Figures la - Ic. As
indicated in Figure lb, the flow separates at X = 0.817 chords. The size of
the recirculation zone is displayed in Figure Ic. The results are all
independent of the time step procedure used to reach the steady state.

In Figure Id convergence histories for this case for two calculations are
shown. The residual displayed in this graph is the root mean square of the
residual of the continuity equation. The calculations were started

348

impulsively by inserting the airfoil into a uniform flow and immediately
enforcing the appropriate boundary conditions. Local time stepping and
enthalpy damping, (see [9]) were employed in each computation; no residual
smoothing was used. For history A the Runge-Kutta scheme with the time step
(At) limitation determined by convection was used; this required choosing a
CFL = 1.0. For curve B a larger Courant-Friedrichs-Lewy (CFL) number was
used by accounting for the diffusion limit on At with the pseudo-time
algorithm. This allowed choosing CFL = 2.5 based on an inviscid
criterion. There is additional work with the pseudo-time scheme.
Nevertheless, the computational time required to reach a satisfactory level of
convergence was reduced by a factor of 1.7.

In the second case we solved for turbulent flow over an NACA 0012 airfoil
with M^ = 0.5, Re^ = 2.89 x 10^, and a == degrees. A 60 x 50 half-plane
grid was used in the computations. The grid spacing at the surface was about
8.5 X 10 chords. The chordwise spacing at the midsection of the airfoil
was about AX = 0.036 chords. Numerical results for this case are presented
in Figures 2a and 2b.

Figure 2c shows two convergence histories for this turbulent flow case.
As in the laminar flow problem, the histories were obtained by computing
without and with the effects on At due to diffusion. The pseudo-time
algorithm was about 1.4 times faster in reaching steady state. This is close
to the factor expected, since we were able to increase the CFL from 1.5 to
2.7, a factor of 1.8. We do not achieve this speedup of 1.8 since there is
some reduction of the effective time step due to the diffusion terms.

349

VI. CONCLUSIONS

The use of the Crocco scheme for a scalar convection-diffusion equation
introduces a scaling of the time step. This reduces the effective time step
so that the viscous stability limit is automatically satisfied. As such the
scheme cannot introduce any fundamental acceleration in reaching the steady
state. The advantage of the scheme is that we do not need to explicitly
account for the viscous time step restriction; it is done automatically. This
can be done efficiently using Runge-Kutta type schemes. In addition, for
variable coefficients or nonuniform meshes this introduces an effective local
time step.

Using this scheme for a system of equations, e.g., Navier-Stokes, has the
additional benefit that a different scaling is chosen for each equation. Thus
each equation has its own appropriate (viscous) time step. This is equivalent
to using a diagonal preconditioning [10] to accelerate the equations to a
steady state. Computations demonstrate that we can gain a factor of between
1.5 and 2 with little programming effort.

We further show that if one uses residual smoothing to increase the time
step then one must also account for the viscous terms. When the smoothing is
applied after the completion of a Runge-Kutta cycle then unconditional
stability is possible only if an odd number of stages is used. Applying the
smoothing after each stage allows for unconditional stability for all
multistage schemes provided a is chosen sufficiently large.

350

REFERENCES

[1] S. Abarbanel and D. Gottlieb, "Optimal time splitting for two- and
three-dimensional Navier-Stokes equations with mixed derivatives." J.
Comput. Phys . 41, 1-33 (1981).

[2] J. S. Allen and S. I. Cheng, "Numerical solutions of the compressible
Navier-Stokes equations for the laminar near wake in supersonic flow."
Phys. Fluids 13. 37-52 (1970).

[3] Yu I. Brailovskaya, "A difference scheme for numerical solution of the
two-dimensional nonstationary Navier-Stokes equations for a compressible
gas." Soviet Phys. Dokl . 10, 107-110 (1965).

[4] L. Crocco, "A suggestion for the numerical solution of the steady
Navier-Stokes equations." AIAA J . 3, 1824-1832 (1965).

[5] A. Jameson, W. Schmidt, and E. Turkel, "Numerical solutions of the Euler
equations by finite volume methods using Runge-Kutta time-stepping
schemes." AIAA paper 81-1259 (1981).

[6] R. Peyret and H. Viviand, "Computation of viscous compressible flows
based on the Navier-Stokes equations." AGARD Report 212 (1975).

[7] T. H. Pulliam and J. L. Steger, "Recent Improvements in efficiency,
accuracy, and convergence for implicit approximate factorization
algorithms," AIAA paper 85-0360 (1985).

351

[8] P. J. Roache, Computational Fluid Dynamics , Hermosa Publishers, 1976.

[9] R. C. Swanson and E. Turkel, "A multistage time-stepping scheme for the
Navier-Stokes equations." AIAA paper 85-0035 (1985).

[10] E. Turkel, "Acceleration to a steady state for the Euler equations."
Numerical Methods for the Euler Equations of Fluid Dynamics , pp. 281-
311, SIAM, Philadelphia, 1985.

[11] E. Turkel, "Algorithms for the Euler and Navier-Stokes equations on
supercomputers." Progress and Supercomputing in Computational Fluid
Dynamics (Edited by E. M. Murman and S. S. Abarbanel), 155-172,
Birkhauser Publishing Co., Boston, 1985.

352

-2.0 p

-1.5-

-1.0

-.5

.5

1.0
1.5

L

1

1

.4 .6

x/c

.8 1.0

Figure la. Surface pressure distribution for laminar flow over an NACA 0012
airfoil (M = 0.5), Re = 5 x 10 , a = degrees).

353

.30

.25

—

.20

—

.15

—

.10

.05

—

—

.05

1

1

.4 .6

x/c

.8 1.0

Figure lb. Skin-friction (based on free-stream conditions) distribution for
laminar flow over an NACA 0012 airfoil (M =0.5 Re = 5 x 10"^

00 ' 00 »

a = degrees).

354

.15

.10

.05

CJ

\

>-

-.05

-.10

1

1

.75 .80 .85 .30

.95 1.00 1.05 1.10 1.15 1.20 1.25 1.30 1-35

X/C

Figure Ic. Streamlines for upper surface at the trailing edge (M = 0.5,

Re •= 5 X 10 , a = degrees).

4
21-

a -2

-4

O -6

-8

-10
-12

A
B

J

1000 2000 3000 4000 5000

Iterations

Figure Id. Convergence histories for laminar airfoil flow calculations.

A — Runge-Kutta scheme without pseudo-time algorithm (CFL number

of 1.0).

B — Runge-Kutta scheme with pseudo-time algorithm (CFL number of

2.5).

356

-2.0 p

-1.5-

-1.0-

-.5

.5

1.0
1.5 L L

.4 .6

x/c

.8 1.0

Figure 2a. Surface pressure distribution for turbulent flow over an NACA 0012
airfoil (M^ = 0.5, Re = 2.89 x 10^, a = degrees).

357

8x10-3

Figure 2b. Skin-friction (based on free-stream conditions) distribution for
turbulent flow over an NACA 0012 airfoil (M = 0.5.

CO '

Re^ = 2.89 X 10^, a = degrees).

358

4
2

2 4 6 8

Iterations

10 X 103

Figure 2c. Convergence histories for turbulent airfoil flow calculations.

A — Runge-Kutta scheme without pseudo-time algorithm (CFL number

of 1.5).

B — Runge-Kutta scheme with pseudo-time algorithm (CFL number of

2.7).

359

CONDITIONS FOR THE CONSTRDCTION OF
MULTI-POINT TOTAL VARIATION DIMINISHING DIFFERENCE SCHEMES

Antony Jameson

and
Peter D. Lax

ABSTRACT

Conditions are derived for the construction of total variation
diminishing difference schemes with multi-point support. These conditions,
which are proved for explicit, implicit, and semi-discrete schemes, correspond
in a general sense to the introduction of upwind biasing.

Princeton University, Mechanical and Aerospace Engineering Department,
Princeton, New Jersey 08544.

New York University, Courant Institute of Mathematical Sciences, New York,
New York 10012.

361

I ntroduction

It is natural that the rapid evolution of increasingly powerful computers
should inspire attempts to solve previously intractable problems by numerical
calculation. One might imagine that within a fairly short time, advances in pro-
cessing speed and memory capacity ought to reduce the simulation of physical
systems governed by partial differential equations to a matter of routine. The
numerical computation of solutions of nonlinear conservation laws has proved, in
fact, to be perhaps unexpectedly difficult. Discontinuities are likely to
appear in the solution, and schemes which are accurate in smooth regions tend to
produce spurious oscillations in the neighborhood of the discontinuities. These
oscillations can be eliminated by the use of strongly dissipative first order
accurate schemes, but these schemes severely degrade the accuracy and usually
produce excessively smeared discontinuities.

The scalar nonlinear conservation law in one space dimension

provides a model which already contains the phenomena of Shockwave formation and
expansion fans. Thus it can be used to provide insight into the likely beha-
vior of numerical approximations to more complex physical systems, while it is
still simple enough to be fairly easily amenable to analysis. A rather complete
mathematical theory of solutions to ( 1) is by now available 11-3).

Equation (1) describes wave propagation at a speed

^<"> = air •

362

The solution is constant along the characteristic lines

3t = ^^""^
provided that they do not intersect to form a shock wave. Tracing the solution

backward along the characteristics, it can be seen that the total variation

du

TV(u) = / ll^l d

ax'

X

is constant prior to the formation of a Shockwave, while it may decrease when
the Shockwave is formed. Corresponding to this property it may be observed that
no new local extrema may be created and that the value of a local minimum is
non-decreasing while the value of a local maximum is non-increasing. It follows
that an initially monotone profile continues to be monotone.

It seems desirable that these properties should be preserved by a numerical
approximation to (1). This will guarantee the exclusion of spurious oscilla-
tions in the numerical solution. Harten (41 has recently introduced the concept
of total variation diminishing (TVD) difference schemes, which have the property
that the discrete total variation

TV(v) =r |v, -v^_,|
k=-«>

of the solution vector v cannot increase. Harten also devised procedures for

constructing both explicit and implicit TVD schemes [4,51.

The purpose of this paper is to state and prove conditions for the construc-
tion of mu It i -point TVD schemes. Conditions are derived for explicit, implicit,
and also semi-discrete operators to be TVD. The conditions are both necessary

363

and sufficient in the case of the explicit and semi-d iscrete schemes. The
reasoning is a modification and extension of the reasoning used by Lax in an
appendix to reference 5. The results were first presented in a lecture at ICASE
in March 1984. The present paper is an amplification and revision of a
Princeton University report issued under the same title in April 1984 161.
In the intervening period Osher and Chakravarthy have given another proof that
conditions (3.12) are sufficient for an explicit scheme to be TVD (7).

364

2. Conditions for Reduction of the i] Norm

One dimensional difference operators act on doubly infinite sequences

u = {^^] > - " < k < =0 . (2.1)

The A] norm of such a vector u is defined as

hl,= I |u^| . (2.2)

•—00

The space of all vectors u with finite i] norm is denoted by £].
A difference operator maps Aj into i] and is of the form

A(u)^ = 1 a^ u . (2.3)

The coefficients aj depend on k, either explicitly or through dependence on u.
In either case we write

a. = a.(k).
J J

Theorem A: The operator A defined by (2.3) satisfies

|A(u)|^ < |u|^ (2.4)

for all u in £ if and only if

I hj^h+j)! < 1 (2.5)

for all h.

An operator A satisfying (2.4) is a contraction .

Proof * The signum function is defined for every real u by

365

ffor u >
for u =
for u <

signum u = { for u = . (2.6)

-1

Now set

s^ = signum A(u)j^; (2.6*)

then, by definition (2.2) of the l] norm and definition (2.6) of signum we have

|A(u)|, = I |A(u)J = I s^ A(u)^ =
k k

h h

where

J

Since S|^ takes on the values ±1 or 0, it follows from (2.7) that

|wj < I |aj.(h+j)| .
It follows therefore from assumption (2.5) that

|wj< 1
for all h. Setting this into (2.7) we deduce that (2.4) holds for all u in Z],

To show the necessity of (2.5) suppose on the contrary that it fails for
some h = hg. Set u equal to

u(0) = j . (2.8)

for il = ho
^ ( for Ji 4= ho

For this u(0) jt follows from (2.3) that

366

and so

|A(u^°^| = i |A(u^°^ I =1 |a (k)( (2.9)

' k '^ k *" %

= I |aj(hy+j)| > 1

since hQ was so chosen that (2.5) is violated. On the other hand it is obvious
from (2.8) that

I (Oh 1
|u |^= 1

This combined with (2.9) shows that (2.4) fails for u(0).

For use in implicit schemes the following result is needed.

Theorem B ; Define the operator B by

B(u)j^ = I b^(k)u^_^. (2.10)

B satisfies

|B(u)|, > |u|, (2.11)

for all u in Jli if

b (h) - I |b (h+j)| > I. (2.12)

j+0 J

An operator B satisfying (2.11) is called an expansion.

Proof ; We define

S|^ = signum u^^ (2.13)

Since |S|^| < 1 ,

|B(u)|^ = I|B(u)J > I s^ B(u)^ . (2.14)

K K

Analogously to (2.7), (2.7)* we have

367

k h

where

J
It follows readily from (2.12) that if u + 0,

\%\> ^ •

w. = y b.(h+j) s.^. . (2.15)*

h ; J h+j

Using (2.13) we get

signum w = signum u

These two imply that

) w^ u, > lul, . (2.16)

f- h h ' ' 1

h

Combining (2.14), (2.15), and (2.16) we get (2.11).

We remark that (2.12) is far from being necessary for B to be expansive.
For example, take the right shift operator T, with

I 1 for j = 1
J ( for j 4= 1

Clearly, T is an isometry:

(Tu)i = |u|i,
but condition (2.12) is utterly violated.

Theorem A has a continuous analogue:

Theorem C; Let u(t) be a di f ferentiable function of t real whose values lie
In iy, and which satifies a differential equation of the form

^=C(u), (2.17)

dt

where C is a difference operator, i.e., an operator of the form

C(u), = I c. u,... (2.18>

368

The coefficients Cj may depend on k and t either directly or through a depen-
dence on u. Then |u(t)|i is a nonincreasing function of t If and only if for
ail h and al I t

c (h) + I |c (h+j)| < 0, (2.19)

Proof ; Define s (t) by

Then

s (t) = signum u (t) . (2.20)

K K

|u(+)|, = i s^(t) u^(t) . (2.21)

K

Since each s^. is piecewise constant.

^|u(t)|i = I s,(t)^ . (2.22)

k

According to equation (2.17),

^ =Ic.(k) u^_. . (2.23)

Setting this into the right in (2.22) we get, after relabeling the index of
summation,

^|u(t)|, -I s^ic (k) u =1 w u (2.24)

k J ^ "^ h

where

"h =\ ^j^^-'J^ Vj • ^2.25)

Suppose Uh + 0; then by (2.20), s^ 4= 0. Multiply (2.25) by s^; using assump-
tion (2.19) we get

^h \ = ^0^^^ ^lo'^j^^'^'^^'hVj "^ ° •

"in u
Since by definition, s^ and Uh have the same sign, it follows that for all h

% % ' 0'

369

this relation clearly holds also when u = o. Setting this into (2.24) we
obta i n

^|u(t)|^ < ;

this proves that |u(t) L decreases as t increases.

Next we indicate why condition (2.19) is necessary. Suppose it is violated
^"^ "^Ot hQ. Let u(t) be that solution of (2.17) whose value at to equals

!1 for k = ho
for k 4= ho
Using (2.23) we get

Summing with respect to k gives

X u, (t_ + e) = 1 + e ) c.(h^ + j) + 0(e ).
f- k . J "^

k J

Since condition (2.19) is violated at to, ho we conclude that for e small enough
positive.

O(e^) .

I u,(tQ + e) > 1

Since

|u(tQ+ e)|, > i; u^(tQ + e)

whi le

|u(to)|, = 1,

this shows that |u(t)|^ is not a decreasing function of t, completing the proof
of Theorem C.

370

3. Construction of Total Variation Diminishing Schemes

Theorems A, B, and C may be used to find conditions on the coefficients of a
difference operator which guarantee that the total variation of a solution does
not increase for

E) expl icit schemes

I ) imp I icit schemes

S) semi-d iscrete schemes.

The total variation of a vector u is

TV(u) = I|u^- V,| .

k

Using the right shift operator T:

^^^\= Vi

we can express TV(u) as

TV(u) = |(l-T)u|^ . (3.1)

We turn now to explicit (2J+1) point schemes

u = D(u ) (3.2)

where

J

D(u) = I d (k)u . (3.3)

K _j J K-j

We assume that the difference operator D preserves constants. In view of (3.3),
this is the case if

I d (k) = 1 (3.4)

J -^

for all k. Schemes (3.3) satisfying this condition can be written in the form

D(u). = \ + I e (k)(u -u ) (3.5)

or In operator notation

371

D = I + E{l-T), (3.6)

where

E = I e. 1-^ . (3.6)*

We want to find conditions which guarantee that D is TVD, i.e., satisfies for
all u

TV(Du) < TV(u). (3.7)

Using formula (3.1) this is the same as

|(l-T)Du|^ < |(l-T)u|^. (3.7)*

Using formula (3.6) we can write

(l-T)D = (l+(l-T)E)(l-T) = A(l-T), (3.8)

where

A = I + (l-T)E. (3.8)*

We now set (3.8) into (3.7)*; denoting

(l-T)u = u*
we obtain the equivalent inequality

|Au*|^ < |u*|j . (3.9)

This is certainly the case if A is an SL] contraction, for which we have derived
In Section 2 the criterion (2.5):

^|a^(h+j)| < 1

J (3.10)

where

(Au), = y a.(k)u. . .
k j J k-j

It follows from (3.8)* that the coefficients a; of A can be expressed in terms
of the coefficients e: of E as

372

0^ ' °0' ' -1
and

a (k) = 1 + e-(k) - e ,(k-l) (3.11)

aj(k) = eAk) - e^_^(k-1), j+O. (3.11)*

It follows from these relations that

L a (h+j) = 1;
J -^
but then (3.10) can hold if and only if for ail j and k

a.(k) > 0.
J

Using (3.11), (3.11)* we can express this condition as follows;

e_^(k-l) > e_2(k-2) > ... >e_j(k-J) > 0,

-eQ(k) > - e^(k+l) >.,.> - ej_^(k+J-1) > 0, (3.12)
1 + e^(k) - e_^(k-1) > 0.

Thus we have proved

Theorem E: The explicit scheme (3.3) is TVD if conditions (3.12) are
satisfied for all k, where ej are the coefficients appearing in formula (3.5)
for D.

We turn next to implicit schemes:

F(u ) = u . (3.13)

Vie take F to be a 2J+1 term difference operator that preserves constants. Such
an F can be written in the form

F = I + G(l-T) (3.14)

where

373

G{u) = y g.(k)u, . (3.14)*

-J<j<J
We want to find conditions under which scheme (3.13) is TVD, i.e., for all u

TV(Fu) > TV(u) . (3.15)

Using formula (3.1) this is the same as

|(l-T)Fu|^ > |(l-T)u|^. (3.15)*

Using formula (3.14) we can write

(l-T)F = (l+(l-T)G)(l-T) = B(l-T) (3.16)

where

B = I + (l-T)G. (3.16)*

We set (3.16) into (3.15)*; denoting

( l-T)u = u*

we obtain the equivalent inequality

|Bu*|^ > |u*|,. (3.17)

This is the case if B is an expansion. In theorem B we have derived criterion

(2.12) that guarantees that an operator B is an expansion:

b (h) > I |b.(h+j) I + 1. (3.18)

j4:0 J

It follows from (3.16)* that the coefficients bj of B can be expressed in
terms of the coefficients gj of G as

bQ(k) = 1 + gQ(k) - g_^(k-1)
and (^-19)

b.(k) = g (k) - g .(k-1), j^O

J J J

Adding up these relations we deduce that

br,(k) = 1 - i b (k+j);
° j+0 J

374

but then (3.18) can hold if and only if for all k and for j +

bj(k) < 0.
Using (3.19) these conditions can be restated as

gQ(k) > g^(k+1) >.., >gj_^(k+J-1) > (3.20)

and

-g_,(k-l) > -g_2(k-2) >...>-g_j(k-J) > 0. (3.20)*
Thus we have proved

Theorem I ; The implicit scheme (3.13) is TVD if conditions (3.20), (3.20)*
are satisfied, where gj are the coefficients of the operator G related by formula
(3.14), (3.14)* to the operator F appearing in (3.13).

We remark that we can combine, as Harten does, theorems I and E to study
implicit-explicit schemes of the form

F(u ) = D(u ). (3.21)

Such a scheme is TVD if F satisfies the conditions of Theorem I and D the con-
ditions of Theorem E.

Finally we turn to semi-d iscrete schemes:

^ = Hu, (3.22)

with H some 2J+1 point difference operator. We assume that u = const is a solu-
tion of (3.22); this is the case if H annihilates all constant vectors. In this
case H can be written in the form

H(u)^ = I "'j(k)(u^_.-u^...,) , (3.23)

-J<j<J
or in operator form

H = M(l-T) . (3.23)*

We want to find conditions on H which guarantee that TV(u) is a decreasing func-
tion of t for all solutions u of (3.22). By formula (3.1), this is the same as

375

|(l-T)u(t) li
being a decreasing function of t. So we multiply (3.22) by (l-T); using (3.23)*
we get

3T(i-T)u = (l-T) M(l-T)u = C(l-T)u (3.24)

at

where

Denoting

C=(I-T)M. (3.25)

( l-T)u=u*

(3.24) becomes

■^ u* = Cu* .
at

According to Theorem C, |u*|i is a decreasing function of t if condition (2.19)
of Section 2 is satisfied:

^n^"^^ ■*■ ^ |c(k+j) I < . (3.26)

° j+0 J

Using (3.25) we can express the coefficients cj in terms of those of M as

fol lows:

c.(k) = m.(k) - m. ,(k-l) . (3.27)

J J J-1

Thus

I c (k+j) = 0;
J ^

It follows from this that (3.26) can hold if and only if

c.(k+j) > 0, j 4= .

Using (3.27) we can restate this as

376

m_.(k-1) > m_2(k-2> >. . .>m_j(k-J) >0 (3.28)

and

-m^iW) > - m^(k+1) >... > mj_^(k+J-l) > . (3.28)*

Thus we have proved

Theorem S ; The semi-discrete scheme (3.22) is TVD if conditions (3.28) and

(3.28)* are satisfied, where the m,- are the coefficients of the operator M
related by formula (3.23)* to the operator H.

377

4. Conclusion

The conservation law (1) describes a right running wave when a(u) is posi-
tive. Conditions (3.12) and (3.28) of Theorems (E) and (S) state that the
explicit and semi-discrete schemes (E) and (S) are TVD if and only if the coef-
ficients of the differences U|^_j - u^-j-i have the same sign as a(u) for j > 0,
(points on the upwind side), and the opposite sign for j < 0, (points on the
downwind side). If the differences are moved over to the right of equation
(3.13), then condition (3.20) of Theorem ('l) states that the implicit scheme (I)
will be TVD if it satisfies a similar condition on the sign of its coefficients.
In all three cases only the differences on the upwind side have the correct sign
for consistency with (1), and can contribute to wave propagation in the correct
direction. In this sense upwind biasing is a necessary feature of explicit TVD
schemes, and it is also useful in the construction of implicit TVD schemes.

It is thus not surprising to find that most of the attempts to design schemes
with the capability of capturing Shockwaves and contact discontinuities, dating
back to the early work of Courant, Isaacson and Rees [81, and Godunov [91, have
introduced upwinding either directly or indirectly. Second order accurate
upwind schemes have been devised by Van Leer [101, Harten [41, [51, Roe [111,
Osher and Chakravarthy 1121, and Sweby 1131. These all use flux limiters to
attain the TVD property.

Another approach to the construction of TVD schemes stems from the obser-
vation that central difference formulas for odd and even derivatives have odd
and even distributions of signs, and they can be superposed and combined with
flux limiters to satisfy conditions (3.12) or (3.28). Upwind biasing is then
produced indirectly by cancellation of terms of opposite sign. One possible

378

starting point for such a construction is a central difference scheme in which
the numerical flux 1/2(fj+i + f ; ) is augmented by a third order dissipative
flux. This scheme is the basis of a method which has been widely used to solve
the Euler equations of compressible flow (141. It can be converted into an
attractively simple TVD scheme by the introduction of flux limiters in the
dissipative terms [151. The modified numerical flux retains a symmetric distri-
bution of terms about the cell boundary j + 1/2. The resulting symmetric scheme
is one of the variants of a class of symmetric TVD schemes recently proposed by
Yee [161. Her derivation follows an entirely different line of reasoning,
building on the work of Davis (171, and Roe [181. In comparison with upwind TVD
schemes, symmetric TVD schemes offer a significant reduction of computational
complexity, while exhibiting comparable shock capturing capabilities.

379

References

1. P.D. Lax, "Hyperbolic Systems of Conservation Laws and the Mathematical
Theory of Shock Waves," SI AM Regional Series on Applied Mathematics, 11,
1973.

2. S.N. Kruzkov, "First Order Quasi-Linear Equations in Several Independent
Variables," Math. USSR SB, 10, 1970, pp. 217-243.

3. O.A. Oleinik, "Discontinuous Solutions of Nonlinear Differential Equations,"
Uspekhi Mat. Nauk., 12, 1957, pp. 3-73, American Math. Soc. Transl., Series 2,
26, pp. 95-172.

4. A. Hapten, "High Resolution Schemes for Hyperbolic Conservation Laws,"
New York University Report DOE/ER 03077-175, 1982, J. Comp. Phys., 49,
1983, pp. 357-393.

5. A. Harten, "On a Class of High Resolution Total Variation Stable Finite
Difference Schemes," New York University Report DOE/ER/03077-176, 1982,
SIAM J. Num. Anal., 21, 1984, pp. 1-21.

6. A. Jameson and P.D. Lax, "Conditions for the Construction of Multi-Point
Total Variation Diminishing Difference Schemes," Princeton University
Report MAE 1650, April 1984.

7. S. Osher and S. Chakravarthy , "Very High Order Accurate TVD Shemes," ICASE
Report 84-44, Sept. 1984.

8. R. Courant, E. Isaacson, and M. Rees, "On the Solution of Nonlinear
Hyperbolic Differential Equations," Comm. Pure Appl. Math., 5, 1952,
pp. 243-255.

9. S.K. Godunov, "A Finite Difference Method for the Numerical Computation
of Discontinuous Solutions of th Equations of Fluid Dynamics," Mat.
Sbornik, 47, 1959, pp. 271-290, translated as JPRS 7225 by U.S. Dept. of
Commerce, 1960.

10. B. Van Leer, "Towards the Ultimate Conservative Difference Scheme. II.
Monotonicity and Conservation Combined in a Second Order Scheme", J.
Comp. Phys., 14, 1974, pp. 361-370.

11. P.L. Roe, "Some Contributions to the Modelling of Discontinuous Flows,"
Proc. AMS/SIAM Seminar on Large Scale Computation in Fluid Mechanics,
San Diego, 1983.

12. S. Osher, and S. Chakravarthy, "High Resolution Schemes and the Entropy
Condition," ICASE Report NASA CR 172218, SIAM J. Num. Analysis, 21,
1984, pp. 955-984.

13. P.K. Sweby, "High Resolution Schemes Using Flux Limiters for Hyperbolic
Conservation Laws," SIAM J. Num. Anal., 21, 1984, pp. 995-1011.

380

14. A. Jameson, "Solution of the Euler Equations by a Multigrid Method,"
Applied Math, and Computation, 13, 1983, pp. 327-356.

15. A. Jameson, "A Non-Oscillatory Shock Capturing Scheme Using Flux Limited
Dissipation," Princeton University Report MAE 1653, April 1984, Lectures in
Applied Mathematics, Vol. 22, Part 1, Large Scale Compuatations in Fluid
Mechanics, edited by B.E. Engquist, S. Osher, and R.C.J. Sommerville, AMS,
1985, pp. 345-370.

16. H.C. Yee, "Generalized Formulation of a Class of Explicit and Implicit TVD
Schemes," NASA TM86775, July 1985.

17. S.F. Davis, "TVD Finite Difference Schemes and Artificial Viscosity," ICASE
Report 84-20, June 1984.

18. P.L. Roe, "Generalized Formulation of TVD Lax-Wendroff Schemes," ICASE
Report 84-53, Oct. 1984.

381

SOME RESULTS ON

UNIFORMLY fflGH ORDER ACCURATE ESSENTIALLY

NON-OSCILLATORY SCHEMES

Ami Harten^

Department of Mathematics, UCLA and School of

Mathematical Sciences, Tel-Aviv University.

Stanley OsherS Bjom Engquist^
Department of Mathematics, UCLA.

and

Sukumar R. Chakravarthy
Rockwell Science Center, Thousand Oaks, Ca.

ABSTRACT

We continue the construction and the analysis of essentially nonosdllatory shock capturing
methods for the approximation of hyperbolic conservation laws. These schemes share many desirable
properties with total variation diminishing schemes, but TVD schemes have at most first order accu-
racy in the sense of truncation error, at extrema of the solution. In this paper we construct an hierar-
chy of uniformly high order accurate approximations of any desired order of accuracy which are

tailored to be essentially nonosdllatory. Tliis means that, for piecewise smooth solutions, the variation
of the numerical approximation is bounded by that of the true solution up to 0{h^~'^), for < R

and h sufficiently small. The design involves an essentially non-oscillatory piecewise polynomial

reconstruction of the solution from its cell averages, time evolution through an approximate solution

of the resulting initial value problem, and averaging of this approximate solution over each cell. To

solve this reconstruction problem we use a new interpolation technique that when applied to piecewise

smooth data gives high-order accuracy whenever the function is smooth but avoids a Gibbs

phenomenon at discontinuities.

^i)Research supported by NSF Grant No. DMS85-03294, ARC Grant No. DAAG29-85-
K-0190, NASA Consortuim Agreement No. NCA2-IR390-403, and NASA Langley Grant
No. NAGl-270.

3B3

I. E^JTRODUCTION

In this paper we consider numerical approximations to weak solutions of the hyperbolic initial
value problem (IVP)

", + /(«)v = = «, + a(uX (1.1a)

u(x,0) = uoix) . (1.1b)

Here u and / are m vectors, and a(u) = df/du is the Jacobian matrix, which is assumed to
have only real eigenvalues and a complete set of linearly independent eigenvectors.

The initial data Uq(x) are assumed to be piecewise-smooth functions that are either periodic or
of compact support.

Let vj = v^(xj,t„), Xj= jh,t„ = na, denote a numerical approximation in conservation form.

vf 1 = v; - X(f^+,^ - /^_,^ = (£, ■ v")^ . (1.2a)

Here Ej, is the numerical solution oiierator, \ = r/h, and /y+L^, the numerical flux, is a
function of 2k vector variables:

//+L'2 = /(v;-i+i-v;+j (1.2b)

which is consistent with (1.1a) in the sense that

/(«,«,...,«)=/(«). (1.2c)

We shall also consider a semi-discrete method of lines approximation to (1.1a) obtained by divid-
ing (1.2a) by t and letting a i

^v, 1 ,. , , ((E. - /) • v),

with /y+y2 again satisfying (1.2.b, c).

The numerical approximation in (1.2) is considered to be an approximation to the cell average of
u:

384

^;^T/J'*""(^.Odx. (1.4)

Accordingly we define its total variation in j: to be:

TV(v'').= TV(v,(-,o) = xiv;.i - v;| (i.s)

J

where | | denotes any norm on R"'.

If the total variation of the numerical solution is uniformly bounded in h, for rs r s T,

TV(v,(-,r) ^ CTV(«o) , (1.6)

then any refinement sequence A - 0, t = 0(h) has a subsequence hj -0 such that

^1
^hj - " (1.7)

where m is a weak solution of (1.1).

K all limit solutions (1.7) of the numerical solution (1.2) satisfy an entropy condition that implies
uniqueness of the I.V.P. (1.1), then the numerical scheme is convergent (see, e.g. [5], [14]).

For the semi-discrete approximation, (1.3), we consider:

VyW == j- 1"" u{x,t)dx . (1.8)

■ The analogous statemraits concerning TV and convergence are valid as well in this case, see,
e.g. [12].

We shall now concentrate on the scalar case, m = \. Extensions to systems will be discussed in
sections HI and V,

Recently total variation diminishing (TVD) schemes have been designed and analyzed [5], [6].
There the approximate solution is required to diminish the total variation (1.5) of the nimierical solu-
tion in time:

TV(v,(-,ri)) ^ TV(v„(-,r2) if ri > rj . (1.9)

385

These schemes trivially satisfy (1.6) with C = 1.

We were able to construct TVD schemes that in the sense of local truncation error are of high-
order accuracy everywhere except at local extrema where they necessarily degenerate to first order
accuracy (see [5], [6], [12], [14], [15], [17]). The perpetual damping of local extrema determines the
cumulative global error of the "high-order TVD schemes" to be OQi^^'^'p) in the L norm. This
improves by one order in steady state calculations, see [1].

In a sequence of papers of which this is the second, we show how to construct essentially non-
oscillatory schemes (ENO) that are uniformly high-order accurate (in the sense of global error for
smooth solutions of (1.1)) to any finite order.

In the first paper [7] we constructed a uniformly second-order accurate scheme which is non-
osdllatory in the sense that the number of local extrema in the numerical solution is non-inaeasing.
Unlike TVD schemes, which also have this property, members of this dass are not required to damp
the values of each local extremum in time, but are allowed occasionally to accentuate a local
extremum.

In this paper the schemes (1.2) are constructed to be essentially non-oscillatory. Our goal is
that, if the initial data uq(x) are piecewise smooth, then for h sufficiently small

TV(v,(-,r -f- Af)) ^ TV(v„(-,r)) + 0{h''^') (1.10)

where N is the order of accuracy of (1.2). This implies that, at each time step, the scheme is non-
oscillatory modulo OQr''^^).

The format of this paper is £is follows. In section n we shall give the design principle and over-
view of the present method, including comparisons with TVD schemes. Section in consists of certain
variants and extensions of the scheme including extensions to systems and to regions with boundaries.
Section TV gives the interpolation algorithm, which is the crux of the method, along with the key
result - Theorem (4.1). Several examples are also given. Section V gives fiuther analysis of the inter-
polation method and an example showing that general non-osdllatory schemes need additional proper-

386

ties (which we believe to be true for the present methods) to guarantee convergence. We also analyze
the truncation error of our methods in this section. Proofs of some technical results are given in an
Appendix. We refer the reader to references [24] and [25] for numerical results using these methods.

n. Design Principle, Overview, and Examples.

In this section we describe how to construct ENO schemes of any desired accuracy.
Integrating the partial differential equation (1.1a) over the computational cell
(x,_V2.*;+y2) X (^. ^«+i). we get

s;^^ = «; - \\fj^^^iu) - fj.^4u)] , (2.1a)

where

/y+L^C") = 7 J^*' f«xj+i'2>t))dt (2.1b)

and

«; = 7 /'*'' u(x,t„)dx . (2.1c)

We shall also be interested in a semi-disaete approximation to (1.1), so we divide (2.1a) by t
and let t 1 0:

d - _ -[/'("(■^/+L'2.0)-/("(^/-L7.0)] ....

ar"-/ h ' ^^-^^^

where again

"/ = T C" "(^'^)'^' • (2.2b)

We observe that although (2.1a) is a relation between cell-averages uf and kJ'*^ the evalua-
tion of the fluxes ^+v2(") ^ (2.1b) requires knowledge of the solution itself, not its ceU averages.
As in Godtmov's sdieme [4] and its second order extensions [20], [2], we derive our scheme as a

387

direct approxiniation to (2.1). We denote by vj* the numerical approximation to the cell averages u?
of the exact solution to (2.1) and set vf to be the cell averages of the initial data. Given v" = {v7},
we compute v""^- as follows:

First we reconstruct u(x,t„) out of its approximate cell-averages {vj"} to the appropriate accu-
racy and denote the result by L(x;v"). Next we solve the IVP;

V, + /(v), = 0, v(a:,0) = L(x;v") (2.3)

and denote its solution by v(x,t). Fmally we obtain vj""^^ by taking cell averages of v(a:,t):

v;^'^ = 7/'*''v(x,t)^. (2.4)

ft </-i/2

The averaging procedure is TVD, as is the exact solution operator. We may conclude, there-
fore, that the design of ENO high order accurate schemes boils down to a problem on the level of
approximation of functions: that of constructing an essentially non-osdllatory high-order accurate
interpolant of a piecewise smooth function from its cell averages.

In section IV we shall construct an essentially non-osdllatory piecewise polynomial of order
^, Q'^i.x'tw) that interpolates a piecewise-smooth function w(x) at the cell interface points:

Q'^(xj+h'2'^) = H^j+vd (2.5a)

and.satisfies, wherever w(x) is smooth

'd''

v'^.

Q^ix ± 0;u) = ^ w(x) + O(frV-i-0, r = 1,...^ . (2.5b)

The key result, contained in Theorem (4.1) in section IV below, is the following. For any piece-
wise smooth function w(x), there exists an h(,> and a function z(x), such that for < h ^ Hq-.

Q^(x;w) = z(x) + 0(fr"'+i) (2.6a)

TV(z) s TV(m') . (2.6b)

388

We shaU use this polynomial together with two different approaches to design ENO schemes.
These methods are:

RP: Reconstruction via the primitive function.

RD: Reconstruction via deconvolution.

We begin with RP. Let W(x) be the primitive function of u(x)

^W = f^ «(*)'iJ • (2.7)

The lower limit shall play no role in what follows, so we choose it to be a = x_y2, for simplicity
of exposition. Thus since we wish to reconstruct u(x) out of its approximate cell averages v, (drop-
ping the r or n dependence) we have an approximation to W(xj+y^

J
Wi^j+i'^ = 2 ViA . (2.8)

i-O

In each cell Ij:{x/xj_y2 ^ x < Xj+y^, Q'^(x;w) is a polynomial of degree N which interpolates
w(xy+y2); i.e., for all j

Q\xj+y2;w) = w(xj^y^ . (2.9)

Thus Q''(x,w) is a continuuous piecewise polynomial, and both of d/dx Q^{x ± 0;>v) are globally
well defined.

Our approximation to (1,1) can be obtained by solving (2.3) with

v{x,w) = d/dxQ^(x;w") = L(_x;v") ,

obtaining v(x,t), 6^ t ^ t and then computing cell averages (2.4). This can be rewritten, using the
divergence theorem, as:

vf 1 = v; - X(^+,^ - ^_,,2) , (2.10)

since

— 1 — = v;

389

because of (2.5a) and (2.8).

Here f?Jr\j2 is computed by averaging the flux function /(«) applied to v(xjj^y2,t) as in
(2.1b).

In the linear case:

M, + AM, = ; (2.11)

this procedure is easy to carry out. The exact solution to IVP (2.3) is

v(;c,f) = L{x - aty) = -^ Q\x - ar.W) ; (2.12)

thus the scheme becomes

v;+i = (E, . v")j = v; - \(^+,,2 - fj-^l) = (2.13a)

given the CFL restriction^^)

\a\\ ^ 1 . (2.13b)

The numerical flux functions fj +y2 defined here involve values of Q''(x;v") for x between
oTy.yj and xj+^^ if a > 0, or Xj+y2 and Xj+y^ if a < 0. Thus, unsurprisingly the resulting
scheme has an upwind bias.

For general /(«) the explicit solution to (2.3) can be difficult to obtain. However, for ^ = 1,
the initial data are piecewise constant:

^^)The restriction can be easily removed in this constant coefficient case.

390

L(x'y) a v;, Xj.y2 ^X< Xj+1'2 •

Thus the scheme becomes:

v;+i = v; - m^xj^i'^ - /(v(^y-L'2)] . (2.14a)

where v{xj^y^ = '^{Xj+v2>^)> for < f ^ t, if the CFL restriction

|X/'(«)|<1, (2.14b)

for all « such that: min(v",vj'+i) ^ « s max(y7,vj'+i), is satisfied.

This is precisely Godimov's scheme [4], which is the canonical three point, upwind, first order
accurate method [9]. Thus our higher order methods are simply generalizations of Godunov's tech-
niques to higher order ENO schemes. The first higher-order TVD (although the concept was not yet
defined) Godimov type method was introduced by van Leer [20]. See [8], [2], and [20] for theoretical
and practical results concerning such TVD methods. The difference here, of course, is that we allow
our interpolant to be arbitrarily high-order accurate even at extrema, and we replace the restrictive
TVD condition [6], [10], by the ENO property.

A key step in this method comes in solving to the Riemaim problem (1.1a, b), with initial data
consisting of two constant states

u{x,G) - «^, a: s

u{x,Q) - uj^yx > .

The unique entropy condition satisfying similarity solution was obtained in [9]. The resulting scheme
(2.14a) can be written:

vf 1 = v; - Xijf^y^ - ff.^^) (2.15a)

where

fjl:.'! = f^(yj,vj^i)

'mn f(u) , if V, s v,+i

maxfiu) , if V; > vj^, ■ (^-^^^^

"/^"^Vi

391

The corresponding semi-discrete approximation is just:

j;^,--W..-i?-.)

(2.16)

Although the high-order explicit method described above can have a complicated flux fimction,
its semi-discrete limit is much simpler. We merely take limits as in (2.2a) and arrive at

-1

f,^] = ir^ i; Q^^^j^^^ - O'^")' j; Q'^"")-^^ + ^'^"""^^

(2.17)

-F

j; Q\xj.^^ - 0;>V), ■£ Q\xj_^.2 + 0;w")\)

i.e., Godunov's method with more accurate constant initial states.

Next we use RD. This time we begin with «(x) and denote by u(x) its mean over
{x - h/2, X + h/2), i.e.,

"W = I l-M "^^"^ = /-L'2 "("^ ■*■ '^^"^ •

(2.18)

Denote by My = u(xj), the cell-averages of u(x).

Again, given cell averages vj which approximate Uj we wish to reconstruct u(x) up to
OQe^*^) in an essentially non-oscillatory way. Here we again begin by constructing a piecewise poly-
nomial interpolant of order N, which we again call Q'^ix;v), that interpolates v at x, for each j:

Q^'ixjiv) = vj . (2.19)

This time C''(j:;v) is a polynomial of degree // in the interval xj ^ x < Xy+i, with possible
jump discontinuities in derivatives at the end points. Then we compute an essential non-osdllatory
piecewise polynomial of degree iV — 1 as follows:

.v-i

1

/•V-^-(x;v) = vy + 2 M^-^jY

(2.20)

m

dx

Q\xj-0',v), f- J?V(x, + 0,v)

dx

392

defined for

^]-V2 ^ ■^ ^ ^y+V2

Here m is the min mod function:

m{x

H

s min([xl,l>'|) if sgnx = sgny = s
otherwise

(2.21)

This gives us our approximation to v, which may have discontinuities at each Xj+y2- We use
this to obtain an approximant to u{x) via a "deconvolution" procedure. We have approximate
derivatives to u(xf^\

'f

u{x)l.,=K'rJ^Q{xj-Q;v),

(f[(2(x; + 0;v)]

+ OQf^\ r = 0,1,...^ - 1

(2.22)

At points of smoothness, we have

ox ,

,V-i-l

.?o r\

(2.23)

i+r

„(,) ^_^ fi - (-TM + Oik-) ,

2'-(r + 1)1 2

for it = 0,1,...,^ - 1-

Thus we may write the ToSplitz upper triangular matrix equation:

^2)This will be shown for piecewise smooth u(x) in section IV.

393

u(xj)

u(xj)

(2.24)

1

1

4-31

1

4-3!
1

u(xj)

This is easily inverted and gives us each of the terms (hd/dxy u{xj), up to OQr').

We replace the left side of (2.24) by the approximations on the corresponding right side of
(2.22) for each v. We invert this system in (2.24) and call the computed approximate values
QCd/dxyvixj).

For Xj.y2 ^ X < Xj+y2f w^ ^te our approximation as

.v-i
L^-\x;v) = 2

v-O

^■k\<^j)

dx

(^ - xjT

K'vl

We need the foUowing:

(2.25)

LEMMA Oi)

The cell average is preserved under this operation, i.e.:

i/;^^^L-v-v,«)^=«

(2.26)

394

Proof

A direct computation gives us:

n f/-w v-(

.v-i

v-O

dx

u(xj)

T'iv + 1)1

1 - (-ir^^

= «/

from the first row of (2.24).

Now we contine our scheme construction as we did using RP. In the RD approach we approxi-
mate (1.1) by solving (2.3) with v(jc,0) = L^~\x;0) = L(x;v") and proceed as above. We again
arrive at (2.10). In the linear case (2.11) the resulting numerical fluxes are defined via

fj+V2 = « Jo' ^•''"K^y+U2 - asr'y)ds , (2.27)

given the CFL restriction (2.13b).

Also the semi-disCTete algorithm for general / obtained via RP in (2.17) is replaced by its
analogue with the numerical flux

fiL^-\xj^y^ - 0), L^-Kxj^y2 + 0)) . (2.28)

m. Variants of the Scheme.

The exact solution to the special initial value problem (2.3) can be difficult to compute. This is,
of course, particularly true when the initial data is a piecewise polynomial of degree higher than zero,
but is also usually true for general systems of equations for piecewise constant initial data, i.e., for
Godunov's method. One can, however, obtain a convergent power series expansion for this solution
see [22], [23].

Godxmov's method is canonical in the dass of (scalar) E schemes, defined in [9]. A consistent
numerical flux yields a semi*disaete E scheme iff

[sgn(«y+i - uj)]fj+v2 =^ [sgn(My+i - uj)]ff+^^ , (3.1)

or equivalently, iff its viscosity is greater than or equal to that of Godunov's method [18]. E schemes

395

are TVD and entropy condition satisfying; tiius tiiey always converge to the correct physical solution
[19], [18]. Examples include the Engquist - Osher scheme and entropy corrections of Roe's scheme -
see, e.g. [3], [16].

One property all E schemes share is tiie fact that they can be obtained by averaging a solution to
a Rieraann problem over each cell, where / is replaced by an approximation /, in equation (1.1) -
see [18], [14]. Thus tiiey retain the ENO property. We may let ff+^^ = f(yj,Vj+{) be any two
point E flux and generalize our semi-discrete algorithm (2.17) to:

i^'--^

^ ^ ^^^""J^^'^ - 05 ^'')' X ^''(^;+v2 + 0; H-")

(3.2)

-f

^ e-Vy-L-a - 0; w"), -^ Q'^ixj..,, + 0; w")

We may generalize (2.28) analogously.
Next we replace the exact numerical flux:

fU:.'2 = jS^f(Hxj.V2>t))dt

by an approxmiation based on a Taylor series as follows.

For Xj_y2 < X < Xjj^y2> we can compute the quantities

'i)'v(^,0)

for V the solution of (2.3), by using a Lax-Wendroff type of procedure.
For example:

|^M) = ^/(v(.,o))

(3.3)

^^f^ = -/■(«(^,0))(«,(x,0))2 -/'(„(;c,0))M„(x,0)

dxdt

396

^^^ = -/■(«(^,0))|-(^,0)-^(;c,0) -/'(«(;c,0)) |^(x,0)

Next we write an approximation to v(x,t):

v^ix,t) = v(x.O) + f ^ (^.v)+-+ ^ ^^^ . (3.4)

Now we replace the integral in (3.3) by a quadrature rule
/ f(yixj+^^; os)ds ~ Ao/(v(x,+L-2,Jo)) +-+ A^ /(yixj+y^^s^)) (3.5)

for s jq < Si'- < jfc ^ 1 .

Finally, we define each value of / above as;

f(yixj+^>2,s,)) = f{v{xj^y^ - 0,j,), v(x^+i^^ + 0,0) (3.6a)

if we base our approach on Godunov's method. More generally we can replace Godunov's flux by its
generalization.

/(V(jf/+ L'2.^r)) = /^(V(*/+ in. - 0. ^r). v(^/+ 1^1 + 0,5,)) . (3.6b)

Thus we approximate (3.3) by a sum of piecewise constant Godunov methods, or approximate
Godunov methods, evaluated at several time layers. TTie quadrature rule, and the value of R, deter-
mine the order of time accuracy of this method.

We note that this approximation need not preserve the essential non-oscillatory property.
Nevertheless, due to the (nonlinear) nature of our ENO interpolant, the method works well numeri-
cally, as is seen from the results in [24] and [25].

Next we consider hyperbolic systems of conservation laws (1.1). In the linear case, /(«) = Au,
where A is a constant matrix with a complete set of right and left eigenvectors r^''), Z^"), correspond-
ing to real eigenvalues X^"), for v = l,...,m. We proceed formally as in (2.1), (2.2), (2.3), and it
just becomes a matter of computing the vector valued function L(x;v") = v(x,0) in (2.3).

397

We decompose an arbitrary m vector w as

m m

W = ^ (/''') • wy': = X W^")

V-i V"l

using the usual /^ inner product. These are used to construct L(x;v") again via the RP or RD
reconstruction approaches.

The RP approach proceeds by computing

i-Q

Then we proceed, as in the scalar case, to compute each of Q'^ix^w^"^), and finally by letting

L(x;v") = 2

r-l

ax

rf") . (3.7)

The RD approach begins by computing Q'^{x;v^^^) which is a piecewise polynomial interpolant of
order N that interpolates v^^) at each Xj. The rest of the reconstruction procedure is done as in the
scalar case, and finally we replace (2.25) by

v-l

For nonlinear systems we denote by Aj = df/du (vj), the Jacobian matrix evaluated at v., and
define Xj"), /j"), and rj") in the usual fashion. This time we decompose

V = 2 (/};) • v) rl = 2 v^^^^ . (3.8)

V~l V— 1

For each v and each Jq, we shall construct an ENO scalar interpolant such that, in the cell
Xj^x<Xj^,^. Q " (x^v'^y) is the unique //th degree polynomial that interpolates v- ^{xj) for
J = 7o> ./o + 1 3nd yy - 1 neighboring points as defined in section IV, and Q " (x.v^")) which
interpolates v " "'{xj) for j = Jq, Jq - 1 and the appropriate N - \ neighboring points.

398

We then construct the m-vector valued ENO piecewise polynomial of degree
N - 1 as follows:

(1-1 v-l H-'

dx

J_
dx

V'^^'^y-O-.vW),

V'''^°^°(xy + 0;v(^))

f ) , (3.9)

for Xj_^^^X<Xj+y2-

We may then deconvolve precisely as in the scalar case and arrive at a vector-valued version of
(2.25). Moreover Lemma (2.1) is still valid.

The RP approach is done analogously.

Thus using either RP or RD we have enough information to compute the vector valued analogue
of (3.2) - the semi-discrete algorithm. This time the canonical method is again Godunov's which uses
the exact solution to the Riemann problem. Other, simpler approximate Riemann solvers may be
used - e.g., Osher's [13], van Leer's for the Euler equations of compressible gas dynamics [21], or
Roe's [15] with an entropy fix as suggested in [16], [17].

The explicit vector-valued construction follows the procedure of (3.5), (3.6), again using perhaps
one of the approximate Riemann solvers to replace Godunov's method.

Various simplifications of these procedures are possible and will be discussed in future papers.

Next we discuss the influence of boundaries on our procedure.

We illustrate the idea by considering the linear equation (2.11), with a ?t 0, to be solved for
t,x> 0, with initial data of compact support. If a > 0, then a physical boundary condition
w(0,r) = ^(f) must be given. If a < 0, then no physical boxmdary conditions are needed.

The modifications needed are two-fold:

(1) At points sufficiently near the boundary our ENO interpolant will lack a choice of least
oscillatory direction. We will choose only among interpolation points which lie inside the region. This

399

procedure has not led to stability problems in oxir niunerical experiments. This can be explained by
the adaptive nature of the stencil in the interior. However, in situations where discontinuities flow
into or out of boimdaries, oscillations may develop. These oscillations do not seem to pollute the solu-
tion globally according to our (now rather extensive) numerical experimentation. We regard this as
essentially the same problem that we have when discontinuities intersect in the interior. We shall dis-
cuss these matters in futiure papers. Some relevant numerical experiments are presented in [24].

(2) Instead of an initial-value problem, at x = we solve an initial-boundary value problem.
This is easy in the scalar case - if a> 0, we just use the given boimdary condition, and if a < 0, we
need no boundary condition since the wave propagates to the left.

For general systems of equations we follow the same procedure, i.e., interpolating in the interior
directions when forced to, and solving an initial-boundary Riemann problem - perhaps approximately.
See [10] for more details about the latter.

One variant of the scheme which we do not recommend involves interpolation of the fluxes to
obtain a high order method. This was draie in [11] in a TVD context, and schemes of arbitrarily high
order away from critical points of the function /(«) in (1.1) were obtained. One might think that
our ENO interpolant might be used on the fluxes using the decomposition of an £" scheme into its
"upwind" and "downwind" parts

'^fi+vi =ff^v2 -f(yj)

<ift^V2 = /(v; + l) - ff+l-2

as in [11]. The difficulty here occurs because of the lack of smoothness of f^ which generally occurs
at sonic points. This degrades the accuracy to be at most third order in L^ at sonic points, if, e.g.,
the Engquist - Osher flux is used, and second order for Godunov's or Roe's methods.

rv. E^ssentially Non-Osdllatory Interpolation and Some Examples

400

Consider a scalar mesh function {vy}j°._„.
We let Q(x;v) be an interpolant:

Qixjiv) = vj = v(xj), j = ...-1,0,1 (4.1)

Xj =jh,h>0.

We shall study a special piecewise polynomial interpolant of degree N, Q'^(x;v), defined recur-
sively as follows:

Definition (4.1)

Q^(x;y) = vj + (x- Xj) ^''^*'~ "^^ , Xj ^ x < Xj^, (4.2a)

= v[Xj] + [X -Xj]v[Xj,Xj+{\ ,

where v[xj^^,...,xj+^] denotes the usual -coefficient in the Newton interpolant.
We also define:

Kg^. = J". ^^ = J + 1 • (4.2b)

Suppose we have defined Q^~^(x,v) for xj ^ x < Xj+-i, and that we also have
^{rin ^'. ^^^''- Then we compute

'^^ = v[^jcW-i)»-'^jc('v-i)+i] (4-2c)

"min "-Bail '■

and proceed inductively.
If lo-^l s \lr\ then

Q\x;v) = Q''-Kx;v) + b" n (x - x^) (4.2d)

401

with

^Sl = ^^„-^' - 1 • (4.2e)

Or if lfl-^1 < \b-\ then

G-^(x;v) = e-^-i(^;v) + c-^ n (a: - x^) (4.2f)

" "mm

with

"max "rnax ^ ^ •

Thus, in each cell xj ^ x < Ay+i, we have constructed a polynomial of degree A^ which inter-
polates v(x) at A^ + 1 consecutive points which include Xj and Xj+i. It is designed so that all its
derivatives are as small in absolute value as is possible, given the above constraints.

Remark (4A)

This interpolant can introduce small oscillations of order fr'^'*'^ even for monotone and smooth
data v(x).

We use the following:

Fxample r4.1'^
Let

v{x) = x^ (4.3a)

N = 2 (4.3b)

Xj = (J- l/2)/i . (4.3c)

The interpolant Q\x;v) for x between x^ and X2 will interpolate v(x) at x^, x^, Xy

402

We rescale, letting

^' = f + I (4.4a)

vW = ^ + Y ■ (4.4b)

We get

Q\x'-y) = -5a:' + 6(x')^ (4.4c)

so a new extrema occurs at x' = 5/12 i.e. at jr = -A/12. The magnitude is OQr') in the unsealed
variables.

Our next result shows that this is the worst possible case for N = 2.

In fact for piecewise smooth data and h sufficiently small the largest possible spurious oscilla-
tions for Q"^ will be OQf'^^).

THEOREM r4.n

For any piecewise smooth v(x), possibly having jump discontinuities, there exist an Aq >
and a fimction z(jc), such that, for all h ^ h^

Q'^(x;v) = zix) + O(fr^^i) (4.5a)

where

TV(z) :s TV(v) (4.5b)

and we repeat:

(2^(x^;v) = v(;c^), j = 0, ±1, ±2,... . (4.5c)

ERQQE

Consider the interval x, ^ x < xj+^ and study two cases;

403

(i) V is smooth in [xj,Xj+{\

(ii) V has a jump discontinuity in [xj, Xj+{\.

Case (i): K v is smooth over the full interval of interpolation [x_(;y), ;c (,/)], standard interpolation

results imply Q"^ = v + OQr''*'^), so we then take z(x) = v{x). Otherwise, for Hq small enough,
there exists an interval containing A^ + 1 consecutive parts such that all divided differences >v[ , , ]
involving points in this interval are bounded independently of h. We call the point at the extreme
right x^y If [xj^-),Xj^^J contains a discontinuity in x, then

mlh'"
where

m = K(^-K + 1 . (4.6b)

rhis follows from the explicit form of v[ ].

Hence the definition of Q'^(x',v) guarantees that, for h small enough, there will be no discon-
tinuity of v{x) in the interval of interpolation [xj^ , ^^cjo ]• ^® result above is still valid:

Case (ii): We may suppose h^ is small enough so that v(x) has only one discontinuity in
[x_(/jr) , x^ ], and it is in [xj, Xj+ 1). For a given interval of interpolation we may decompose:

RBfl tDMX

V = w + H (4.7)

where w is Lipsdiitz continuous and H is piecewise constant with a single jump which occurs in

We have in [xj, Xj+{):

Q^ixiv) = Q^ix^w) + Q\x'^) , (4.8a)

404

where:

f:(N)

-nn "mix u-y+i

(4.8b)

and

"mm

"■mix

(4.8c)

(where C always denotes any universal positive constant).
This implies that

^ Q'^ix^w)

C .

(4.9)

By Rolle's Theorem the interpolant Q^{x)H) of the piecewise constant function must have an
extremum in every interval (x„, x^+^ for v * j, ATW s r ^ AT^. This makes a total oi N - 1
extrema. Since the interpolant is of degree N, it must be monotone in [xj,Xj^{\.

Thxis, for A = 1, we have

max

dx

Q\xm

OO

(4.10a)

For general h, the scaling gives

max

dx

Q\x;H)

h

(4.10b)

Thus (4.8)-(4.10) imply that Q'^(x;v) is monotone in [xj, Xj+{\. We take

2(x) = C'''(^;v) .

(4.11)

On the interval [xj,xj^-J

TViQ\x;v)) = \vj^, - vj\ s TV(v)

(4.12)

The theorem is proven.

405

We also have

Remark (4.2)

Let v(x) be piecewise polynomial of degree ^ N. Then in any interval [xj,Xj+{\ in which
v(j:) is not discontinuous the interpolant is exact

Q-'ix-fi) = v(x) .

We now compute "second" and "third" order accurate approximations to the linear problem.

u, = -u, (4.13)

Using RP for N = 2, we have, for ^y.^^ ^ ^ '< ^j+v2 >

Q\x',w) = wj_^^ +ix- xj.y^vj + (4.14a)

+ iiL_ii^L^i±i^ ^(A_v^, A.V,)

where

and

m(x,y) =x a \x\^\y\ (4.14b)

mix,y) = y if \y\>\x\

^^VJ = T(yj^i - vj) . (4.14c)

The algorithm becomes

vj+i = v« _ xA_[v7 + p^Y^ «(A_v7, A+v;)] . (4.15)

This is a TVD scheme [6] for X < 1 which is second order accurate with a first order degen-
eracy at critical points.

For N = 2 with the RD approach, a simple calculation gives the algorithm for |X| < 1:

406

l-\

1 -/

vj+i = v;-XA_[v;+ -—^ m[A_vy"+-i-m(A_A^v;, A.A.vj")

(4.16)

1 -.

A^v; - ^m(A.A.v;, A_A+v;)]]

(In [7] we obtained a similar algorithm, with both of the m replaced by m. We proved that the
scheme in [7] was truly non-oscillatory.)

The sdieme (4.6) is truly second order accurate, even at critical points and converges for
|\| < 1, at least according to extensive numerical tests.

Using N = 3 in the RP approach gives us for jc^.^j ^ x < Xj+y2'-

If |A_Vy| s |A+Vy| , then:

(4.17a)

2^(x;>v) = >v,_,, + (. - x,.,^v, + l (^-^/-L^(^-^/^i.^ ^_,^

+ —(x - xj.^^ix - xy-y^(x - Xj+^^mC^.A.Vj, A.A^v,)

If |A_Vyl > lA+Vyl , then

(4.18b)

Q\x;w) = Wj.y2 + (^ - X]+y^V] + T- ^ r ^ A+Vy

+ — (j: - a7_v2)(^ - ^y+L'2)(^ " Xj+-i,.^m{L.L^Vj, A+A+Vy) .

Then our numerical scheme becomes for |X| < 1:

v;*i = v; - xA_[v; + ^^-- ^ m(A_v;, A+v;)

1 - X

(4.19)

■i- (X - 1)(X - 2)m(A_A_v;, A_A+v;), if |A_v;| :s |A^v;|
-i- (X - 1)(X + l)m(A_A+v;, A+A+v;) , if |A_v;| > lA^v;|

A07

This scheme is third order accurate except perhaps at points where u^ or u^^ = 0, at which it
may degenerate to second order accuracy.

For iV = 3 using RD we have for xj ^ x < Xj+^:
Q\x;v) = vj (4.20)

(x—Xi) (x—xMx—Xi+'i)

6A3

2A2

(x-Xy_i)(j:-j:y)(j:-x,+l)m(A_A_A+Vy,A_A+A+v,),if|A_A+Vy|:s|A+A+Vy|

^x-xj){x-XJ^{){x-Xj^■^mi^_^^^^Vj,^^^J^^+vj),^i\^_^^Vj\>\^J^^^Vj\

We can derive a globally third order accurate scheme by using (2.20), (2.24), (2.25), and (2.27).
We omit the details here.

It should be stressed that our algorithms are to be obtained recursively using the computer. We
have written down a few numerical fluxes here just to give the reader some idea of what they look
like.

V. FURTHER THEORETICAL RESULTS AND EXAMPLES

While Theorem (4.1) is encouraging in that it shows us that the interpolant (2^ is indeed essen-
tially nonosdllatory, more analysis needs to be done. The schemes designed in section II do not use
this function in a simple enough fashion for us to prove the desired estimate (1.10), even if v;,(x,r) is
piecewise smooth.

As a step in this direction we consider the method based on RP applied to a piecewise continuous
function. A canonical example involves the interpolant Q'^{x;g), where g{x) is the primitive of a
Heaviside function normalized to be:

g{x) = a. - x,x -^ fx (5.1a)

408

six) = 0, X > a (5.1b)

for ^ a < 1.

We let h = 1 and compute the least oscillatory piecewise polynomial Q^{x,g) which interpo-
lates g{x) atx = j for each integer ;. By Remark (4.2) we have

Q^ix-^g) " 8(x) for X ^ and ;c s 1 . (5.2)

We need only compute Q'^(x;g) for < jc < 1. We wish the reconstructed function
d/dx Q'^(x,g) to be a non-oscillatory approximation to g(x). This reduces to showing that on
0^ jr ^ 1

-1 ^ ^ Q^'ixig) s (5.3a)

-^ ^V;^) ^ . (5.3b)

The least oscillatory polynomial on the interval ^ x s 1 will be one of the N + 1 polynomi-
als of degree N, Qx(x;g), which interpolates g(x) at the A^ + 1 consecutive points

{K -N,K -N + 1,...,0, 1,...,^}

for l^K^N.

■ In the proof of Theorem (4.1) we showed that any polynomial which interpolates the derivature
of this function g'(x) through these N + 1 points is monotone on the interval ^ x ^ 1 In con-
trast we have

Example (5 A')

thus:

riN - f ^ _L {x + N — iMx + N — 2)..j: mi fe a \

Q\ = {o-- x) + J ^^ ' [1 - a] (5.4a)

^ Ql ix,gX . , = -1 + (1 - a)[l + ... -f 1.] > .

409

for N(a) sufficiently large, when a is fixed: 1 > a > 0.

Thus, in order for the inequalities (5.3) to be valid, we need the special properties of the least
oscillatory interpolant of g(x). We have:

I^emma (5.1^

The least oscillatory polynomial of degree N is Ql[ iff
1- K/N:Sa<l-(K- 1)/N, K = 1,2,...//.

Finally we have:

T^mma r5.2'>

The polynomial obtained in the statement of Lemma (5.1) satisfies the inequalities (5.3).

We shall present the proofs of these claims in the Appendix.

Next we consider the method based. on RP applied to a smooth perturbation of a liaviside func-
tion g'(x). We find here two new problems.

(1) The error between d/dx Q'^ix^g) and g'{x) in the cell next to the interval containing the
discontinuity need not be OQt') - it can be as bad as 0{K) for N > \.

(2) The variation in this ceD can increase- i.e., Wax[d/dx Q'^(x',g)\ in this cell can exceed that of
g'(x)-m this cell by 0{h^) for N > 2.

On the plus side we note that these are somewhat pathological examples, that the error and
growth in variation are indeed decaying with h, and that two cells away from the discontinuity all
seems well in that the error and possible variation growth appear to be OQt'). Nevertheless we
expect to investigate other ENO interpolation procedures as well as alternative reconstruction tech-
niques, with an aim towards removing these (hopefully minor) problems.

410

Example ^5.2)
Let

8(x) = ^^L±Mt + a(x + Bh), X > -Bh (5.4a)

g{x) = -X - Bh,x rs -Bh for 1 > 5 > . (5.4b)

Then the function we are approximating, g'(x), satisfies

g'(x) = -l,x:s-Bh (5.5a)

g'ix) = x + Bh + a x> -Bh . (5.5b)

We shall obtain Q'^ix^g) which interpolates g(x) at grid points xj = jh, j = 0,± 1. We are
interested in Q'' for :s x ^ h. We shall arrange a and fl so that

T- Q'^'ixig) for ^ = 2 and 3
ax

both have OQi) pointwise error compared to that of g'(x) on this interval.

We do this as follows:

For O^x^ h:

QKx;g) = giO) + j^igih) - g(0)) .

Next we arrange a and 5 so that the three consecutive points i-h,g(~h)), (0,(^(0)), and
ih,g(h)) are collinear:

gih) - 2g(0) + g(-h) = (5.6a)

or

gi-h) = A2 - (B Ifh^ _ ^^g _ j^ ^ ^^j _ ^^ ^^^^^

2

or

411

|(5 - 1)2 + (a + 1)(5 - 1) - A =

(5.6c)

We solve this obtaining

B(h,a) = 1 + ^- + 0(A2) ,
a + 1

(5.6d)

and since we want ^ fl s 1, we take a < -1.

Thm we have for ^ x ^ h

which interpolates g(x) at j: = -h,0,h and

which dearly differs from g'(x) by 0(h) at some points in this interval.

We also claim that d/dx Q\x,g) - g'(x) is OQi) in this interval as well. It is easy to see that
Q^ will be chosen to interpolate g(x) at.jc = -h,0,h, and 2h. Thus in our interval of interest:

Q'{x;g) = QKx',g) + (^ - ^>(^ + h)

6h^

^^ -^^^^ + ah(B - 1) + h(B - 1)

Thus

5 (I'M = f (B - 1)

^^^+(1+=)

and then

J^^^fc3(,^.) = 1(5-1)

^^ ~ ^^^ + (1 + A)

= f + 0(A2)

and the error is still 0(h) since varQ^^j;;, g'(x) = h.

Our next example well allow for an increase in variation in this cell, although it will still decay
with h as A i 0. Let:

412

gix) = ^"^ "^^^^^^ + b(x + Bh), X > -Bh (5.7a)

g(x) = -X - Bh, X IS -Bh (5.7b)

for 1 > 5 > .

Then the function we approximate is:

g'(x)''-Ux^-Bh (5.8a)

8'{x) = ^L±Mt + b . (5.8b)

This time we want the points {~h,g{-h)), (^,g((S)), (h,g(h)), and (2A, g(2A)) to all lie on
the same parabola. This means that

= h^ + h^ ^^ " ^^^ + (1 + b)h{B - 1) (5.9a)

or

5 = 1-^ (5.9b)

Thus we take b > —1.

On the usual interval s j ^ /« we have

Q\x;g) = 5(0) + I (if(/i) - gm (5.10a)

Q'(^;«) = Q\x',g) + ^^y/^ k(/.) - 2g(0) + g(-A)] (5.10b)

and by (5.9):

Q\x;g) = Q^(x;5) (5.10c)

The function g'{x) is monotone on the interval s x ^ A as long as i is not OQi^) so:

A13

vai g'(x) = ^h\l + 2B)

while

var f Q^(x.g) = i^(^) " ^ ) ^ ^(0)1 = h\l + B)
OsisA dx h ^ '

Thus an oscillation of order hr/l is induced in what should be a third order method.

We note that the discontinuity in g'{x) is rigged so that it occurs at a distance OQP) from a
grid point. This is a bit pathological, but is certainly possible.

This oscillation is maintained even when we increase the order. For example, in the same inter-
val it can be easily shown:

Q*ix;8) = Q\x',g) + ^^ " ^^^" : '^^^ ^ '^ [Ei-h) - 8(-h)]

2Ah

where g(x) is the continuation of the cubic polynomial g(x) to x negative.
Thus

g(-h) - g(-h) = h(l - 5) -

h^

(B - 1)3 + bh(B - 1)

= h(l + 6)(1 -B) + ^(1-By = h^ + OQi^^
6

A simple calculation gives us

OSvsA dx

I-

+ 0(h') ,

and we again have a variation increase 0(h^) in this interval.

Next we show that a scheme, which is non-osdllatory for relevant data in that new extrema do
not develop on the initial data as h inaeases, can still be extremely unstable.

Example 5.4

414

by:

We take as initial data

We approximate

U, = -K, (5.11)

VJ--1 = v; - XA_m(v;+i,v;) . (5.12)

v]'-0,;s-l (5.13a)

v§ = a (5.13b)

V? = e - a (5.13c)

vO-O.yaa (5.13d)

for < € « a, < X < 1/2.

An explicit computation gives us:

v/«-vj' if j^ -l,ja2 (5.14a)

va = fl(l + X) - Xt (5.14b)

v.i = (e - a)(l + X) . (5.14c)

Thus the "shape" of the initial data is invariant in time and

vg -00

V? -1 —00 .

Now we analyze the truncation error TE for our two methods. We begin with RP applied to the
linear equation (2.11) and arrive at (2.13). In this case a precise expression for TE is:

TE= ^ ^-[Q\xJ^v2^y^)-W{xJ^^,^ - (5.15a)

415

- o-v

Q\X]+vi - aT;^) + W(xj+^^-a7)]

We recall

W(x) = f u(s)ds with

(5.15b)

and

n^j+i'd = Q\xj+v2l^)

(5.15c)

dx

Q(':w) - ^

^{x) = 0(/r^+^-^)

(5.15d)

in regions where ^(a:) is sufficiently smooth.

It is clear that the TE is 0{\v^) as long as the coefficient multiplying the A-'^+i-^ termisdif-
ferentiable when for v = 1. This will be true in general if the stencil of points used for the interpo-
lant in two consecutive intervals is invariant under translations. This is true in smooth regions if none
of the derivatives of «(a;) up to order N_- 1 vanish in a neighborhood of this interval.

We thus have

Theorem C^.W

TE for the explicit and semi-disaete methods based on RP approximating a linear equation is of
order

TE = 0(frV), if

dx

u{x) ?t 0, r = 1,2,...^ - 1

(5.16a)

TE = O(fr^-i) otherwise

(5.16b)

For the full nonlinear problems the algorithm (2.14) can easily be shown to satisfy estimate
(5.16b) above.

The computational evidence is that (5.16a) is valid under conditions stated there for the non-

416

linear case. We believe this to be true, but do not prove it here.
Next we state:

Theorem (fi.2\

TE for the explicit and semi-discrete methods based on RP for general nonlinear equations is at

least

TE = 0(fr''-')

(5.17)

Finally we analyze HE based on RD. Recall we are given via interpolation the values:

a^ = h^id/dxy u(xj) + 0(k^*^), v = 0,1,...^ - 1

Next we compute

K = K'

dx

u(xj) + OQr')

using the matrix equality:

Ofl

'^.v-i

1 a, • •
1 tti
0- ••

a.v-i

1

u(xj)

*i

.V-l

u(xj)

(5.18a)

417

u(xj)

a.v

Si

+ C?(fr^*i)

where

_ 1 + (-1Y

n = J i_

2^(V + 1)1

(5.18b)

Call C the upper triangular Toeplitz matrix on the right above. We approximately invert the
system, obtaining

■ *o'

' °0 '

•

= c-i

•

=

*.V-1_

fly-i

u(xj)

t \

u(Xj)

/ \

dx

y.-i

u(xj)

(5.19)

+ c-i fr^

«(Xy)

«!

+ 0(/rV+i)

Next we compute the function L^~\x',u) as in (2.25), for J[/_y2 ^ J^ < ^y+v2

,v-i

i:-^-'(^;") = 2 *v

(^ - xjT

r-O A^vl

(5.18)

= <x) -^

.V

uixj)

- T.VV

A^lfrV

(X - Xj}

418

ax

N

u(xj)

1,

"-l

+ 0(frV+i)

To show that TE = OQt^) in this case, we need only prove that

D<i\L^-\x',u) - u(x)] = 0(frV)

(5.19)

for u smooth. This follows by the smoothness in both Xj and x up to OQr^'*'^) of the remaining
terms on the right side of (5.18).

Thxis we have:

Theorem 5.2

TE for the explicit and semi-disaete methods based on RD approximation for general systems of
equations is OiJr').

We also note:

Remark (5.1)

We have been imable, so far, to prove that these methods are indeed essentially non-osdllatory
although our present results show that the interpolation upon which the whole framework is based
does indeed have this property.

Remark (5.7\

If uq(x) has two neighboring discontinuities and h is not sufficiently small, our present

419

methods, for N > 2, can result in nontrivial spurious overshoots. We shall remedy this difficulty in
subsequent papers.

Appendix

We shall provide the (lengthy) details of the proofs of Lemmas (5.1) and (5.2).

Proof of I^mma r5.n

We shall use induction on N. The result is trivially true for N = 1. Suppose it is true up to
N. We consider the interval

1 -

N + 1

a< 1 -

^-1

,A^+ 1,

We divide this into two parts

Ir- 1-

K-l

^ a< 1 -

K - 1

< 1 -

K -2

N

N+lj

N

•

K

:S a< 1 -

f \

K- 1

N + 1

N

after verifying

1 -

\k-1]^. (k-2]

N+1 ^^

N

(Al.a)

(Al.b)

and

K -2<N

1- Tr^<l -
N+1

K- 1

N

K-\ ^ K

N N + 1

420

N/K - ^ + X" - 1< -^

A.

K - KN

Thus by the induction hypothesis: for a € I^, Q^{x',g) = Q'f:-^X^',g), (if AT = 1, Ij^ is empty),
and for a € 7^, Q''{x;g) = Q^{x;8).

We wish to show that for a € /.j U /£, that Q'^'*'''-(x;g) = Q'^*Hx',8)- Using the iterative defini-
tion of Q^'^'(x;g), we must compare the t\vo Newton coefficients

-R = ia-K)-

(N+1)
1

(a-(A:-l)) + - + (-l)^-i

1)

(A2.a)

-S = (a-iK-l))- r ^M (.a-(K-2)) +

4....-f(- 1)^-2

(^!;](^-

1)

(A2.b)

We wish to show

\R\^\S\

(A3)

for these values of a.

To prove this we need the following:

F^ct fAl'i

n

n

+ - + (-1)

-1^^

n

= (-1)^

n — 1
K

for :s a: ^ n - 1

Fact rA2-)

421

K

for 1:SK ^ n - X.

-(K-1)

l\

+ ... + (-1)^-1

n
AT- 1

= r-n^-i

(-1)

(n-2)
K- 1

Proof of Fact A1

Again we do it by induction. It is true for K = 0. Suppose it is true for K. Add
(-1)^+1 ^ ^ J to both sides of the equality. On the right we have

i

il

(n-

1)1

^(n-K- l)l(K + 1)! (n-K- 1)IKI J

_/-_lN^+i( (n-l)l ]
^ ^ [(n-^ -2)1(^ + 1)1

n

(K+1) ]

n-K-1

in- K-1)}

= (-1)^+1

K + 1

Proof of Fart A?,

Using induction. We see that it is true for K = 1. Suppose it is true for K. Then

(K+1)

^ /

- K

+ ••• + (

-)^t)-H-

-(K-1)

M

+ ... + (-1)

-n^-i

n

(by Fact (Al)),

(by the induction hypothesis)

1)^

n
K

422

= (-1)

-U^

(n - 1)1

(" - 2)!

[Kl(n -1- K)\ {K- l)!(n - I - K)\

'^ ^ ^!(n -2-K)\

n - 1

K

[{n-\-K) in-K-K)]

= (-1)^

K

Using these facts, we have:

/? = (!- a)(- 1)^-1

K- 1

-1^^-2

+ (-1)

K -2

(A4.a)

5 = (1 - a)(-l)^-2

f ^ 1 .

1>

,jr-3

fA^-ll
^-3

(A4.b)

We note

(-l)^-2i? =

a:-2

- (1 - a)

' N
K-1

(ff-l]_( N ](k-i]

K-2

(1 - 1) =

so

|i?l

= (^:J]-a-»)

r ^ 1

AT- 1

Also

(-1)^-25 = (1 - a)

r N \

K-2

-rA

423

K- 1

N + 1

' N '
K-1

K )

-

'N-l]
K-3

'N-\\
K-3

[ NiK - 1)

- 1

=

(N-l)
K-3

N - K + 1

UA^+1)(^-2)

[iN + l)iK-l)}

(1-a)

' AT ^
K-1

—

{N - r

K-3

We check

|5| ^ \R\

(1-a)

m

Nl

{K - 2)[(N -K + 2)\ {K- 1)\{N - K + \)\

(N - ni

(N - 1)1

{K - 2)\iN -K + l)\ {K- 3)\{N - K + 2)1

^Yrf'mK -l) + NiN-K'+ 2)]

(K - 1)(N -K + 2) + (K - 1)(K - 2)

N^N

For 4 the two Newton coefficients are: the same R and

5 = (1 - a)(-l)^

K

+ (-1)^-1

K- 1

This time we have

(-l)^-ii? = (1 - a)

r fi \

K- 1

K-2

K - 1

N

{ N \ {N-l\

Ar-iJ-[i^-2j = o

and

424

(-1)^-^5 =

K - 1

M

f/v-n

-(1-

ct)

K

s

K- 1

;y + 1

K

Finally we check

K- 1

1 -

N

N + 1

>0

|5| S \R\

or

(N - ni

(N-l]

(N-l]

K - 1

+

K -2

(1-a)

r A/ ^

X"- 1

+

'N'
K

{K - \)\{N - K)

1 +

K- 1

N -K+\

KN (N - Di

A/ + 1 (K - 1)I(A/ - a:)!

(1 - X- + 1)

N

^- Ar + 1 A^ + 1

Thus Lemma (5.1) is proven.

N + 1

K(N -K + 1)

Proof of Lemma (5,2)

We start with a general geometric resiilt.

Fact rA.3^
Given

-^ Q^(i) ^

± QKo) ^ -1

(A5.a)
(A5.b)

425

f ei^(i) ^ -^ »)

(A5.C)

Then

Ql{x) a for :s x s 1

(A6)

Proof nf Fact A.1

RoUe's Theorem tells us that d/dr Gj^(j:,g) = at least once in each interval

(1,2) {K - 1,K) and d/dx Q^(x,g) = -1 at least once in each of

(K-N,K-N+ !),...,(- 2,-1), (-1,0). Tf K = l, this means that d^/dx^ Qlix^g) = at least
N -1 times for a: < 0. Thus d/dx Q^x,s) is monotone for rs x < 1. If K = N, then a simi-
lar argument shows that d^/dx^ Q^ixig) = at least ^ - 2 times f or j: > 1 and the same monoti-
dty result follows. Given (A5(c)), this takes care of these two cases.

For 1< K < N we proceed as follows. If d/dx Gj^ = at least once in addition to these
values mentioned above for l^x :£ K, then it equals at least K times for ;c s 1 and -1 at
least N - K time for x s 0. By our usual argument this means that it is monotone on (0,1), and
we are finished. Similarly if d/dx Q'^x) = -1 at an additional point for K - N ^ x ^ 0, the same
conclusion follows.

If both of these possibilities are false, then the graph of Q'f:(x;g) looks like for 1 s x:

(a) Kodd

K - 1 K

(b) K even

426

Fig. Al: Q^ for x > 1

and for K - N s x ^ 0, the graph looks like:

(a) N - K odd

K - N

(b) N - K even

K - N -2-1

Fig. A2: C-;^ for j: s

K the leading coeffident of Q^ vanishes, then we have a polynomial of degree iV — 1 , and its
derivative is monotone on (0,1) as per our usual argiunent.
Otherwise we consider the following cases.

Case (1) K and N — K even. Then if the leading coeffident is positive it follows from glancing at
Rg (A2.b) that d/dx C)r = -1 for some x < K - N and we are finished. K the coeffident is nega-
tive then Hg. (Al.b) shows us that d/dx i3)r = for some x> K and we are again finished.

427

Case (l) K and N - K odd. Then if the coefficient is positive Fig (Al.a) shows d/dx Q^ = for
some ;c > a:. K the coefficient is negative then Fig (A2.a) shows d/dx Q^= -1 for some
x<K - N.

Case (3) K odd, N - AT even. K the coefficient is positive then Fig (Al.a) shows d/dx Q^ vanishes
for some ;c > AT. . If the coefficient is negative then Fig. (A2.b) gives us the desired result.

Case (4) K even, N - K odd. If the coefficient is positive then Fig. (Al.a) gives us the desired
result. K the coefficient is negative then Fig (Alb) does it.

To prove Lemma (5.2) we need only verify the inequalities (A5). We finally write down the
formula for Q^:

Lemma (Al)

Gat- a x + j^^ (AT - AT + ;)!

(-:y-.(i-«)f'-^^r']

(A.7)

-lV-2

+ (-iy

(N ~K + j-

2)

N - K

■

where we define

B

= if either 5 < or 5 > A.

EroQf:

Qearly Q^ = a - x for x = K -N,K - N + 1,...,0.
For X = 1,2,. ..,Jt, we need

428

a -;c + 2

'N - K + x')

(1 - a)(-iy-i

'N -K + j - 1^

(A.8)

^^-^rf-lT'^

=

TTiis will follow if, for all integers M & 0:

Fact rA.41

1=2

Af + V
M + j

(-ly

-1

'Af + ;• - r

val

Fact (A.5):

v=E

/-I

M+j + lJ^ ^^ [ A/

va: 1

Proof of Facts (AA^ and (A.S>

We shall again use induction: For A/ = we need

0= -

E(-iy

-1

;-i

V

- 1

= 2 (-ly

;-o

V

This follows from Fact (A.1) for v = n = K + 1.
We also need

v=E

y-i

'v + l

; + i

(-ly

-1

(A-9a)

(A.9b)

(A. 10)

(A.11)

2

J— I

; + i

(_iy-i_

fv + l]

(v + l]

+

1

by (A.10).

Suppose both Facts are true up to A/. For A/ + 1 we need

429

1=2

/-I

(M + l + v)

(-y-'pl)

(A.12)

= V (K±1±Jl\ (m + i + l ]
jif, (m + i+j}[ M + 1 J

M + j

(-ly

-1

(M + j- r
I ^ J

1

■y?i M+1

'M +1 + v]
M + 1 +jj(

ly

-1

M

— V + 1 w , I = 1 (by the induction hypothesis)

Now we show (A.9b) is true f or M + 1 by induction on v. It is dearly true for v = 1. If it
is true for v — 1, we consider

Ai^+^r ly [ M J

2

^-1

(M + v + i\

M + j + 1

i-irfT']

+ (-irn M

i[::;.]["Tyy

•(V - 1) + 2

or

v=2

y-i

(M + v + i

M + ;• + 1

(-v-r^ri

Now we verify (A5.a)

-I: (2)^(0) = -1 + i (-ly-i J^-^)l

c2r

y-i

(A/-if + ;)I

430

(1 - a)(-iy

-iv-if^-^^_Y-ii

^i-^-f'J^-''

a -1

or

or

» - «'! [w^ - 1 (s-k4(^->c^J-» ^ '

(A. 13)

We use the identity: for A < 5

3 J _ ^ ( 1

1 1

J) A-l B

(A.14)

So (A 13) becomes

-a i ! + (//-/:)

' -41^0

//-^ N

■^ 1

J-N-K+l J)

1

.V

1 -

«■- 1

N

.V

a 2 -t(^ - a- + 1) - a- .

J-N-K+l J

If we replace the right side above by ^ + 1, we get

2 ^{N -K)-K-l

J'N-K J

= 2 ^(^-AT + i)-/:- 2 -

431

Thus the right side above is decreasing with K, and we need only verify the inequality for

K = 1

2=^^-1 =
M

Next we compute:

dx

^;^(1) = -1 +

2 -

(. v-l V

(1-a)

y-2

{N-K + j)\

(1 - a)(-iy-i

'N - K + j -1^

+ (-iy-2

'N - K + j -i
J-2

Rearranging terms and simplifying gives us:

(-a)

■"'4"' 1 4 J N-K + l ]

y-l

(A,15)

2

(N-K+l)

jf^ iN-K + j-l)iN-K + j)

1 .

Now

;-2 (j-im-K+j) yt-zj-l j^2N-K+j

The first term in (A. 15) this becomes

(1 - «) E 7

Using the identity (A. 14), the second term becomes:

(N-K+l)

N ~ K + 1

1_
N

= 1 - 1 +

K - 1 _ K -1

N

N

432

So we have to check:

^^-^It^^^^

^y 1 + JLzJ.^l

Again, if we replace AT by ^ + 1 on the left side above, it inaeases by 1/N 2/-;r+i 1/j. ITius
v/e need only verify:

N N N

1 :s 1
The last step is to verify that:

or

± &) ^ ± QliX)

y-,v-jc+i J N j~ic J ^

or

1 '^1 'V 1

^^(i-a)27 + « 2 7

<V-1 , ( N-l 1 ,V-1 ,

0^27 + ct 2 7-2-7

•V-l 1 ,V-1

"ITiiis Lemma (5.2) is proven.

433

BIBLIOGRAPHY

[1] S.R. Chakravarthy and S. Osher, "A new class of High Accuracy Total Variation Diminishing
Schemes for Hyperbolic Conservation Laws," AIAA paper #85-0363.

[2] P. ColeUa and P.R. Woodward, "The piecewise-parabolic method (PPM) for'gas-dynamical simu-
lations," J. Comp. Phys., v. 54 (1984), 174-20L

[3] B. Engquist and S. Osher, "Stable and entropy condition satisfying approximations for transverse
flow calculations," Math Comp, v. 34, (1980) pp. 45-75.

[4] S.K. Godunov, "A finite difference method for the numerical computation of discontinuous solu-
tions of the equations to fluid dynamics," Mat. Sb., 47 (1959), pp. 271-290.

[5] A. Harten, "On a class of High Resolution Total- Variation-Stable Fmite-Difference Schemes,"
SINUM, v. 21, pp. 1-23 (1984).

[6] A, Harten, "High resolution schemes for hyperbolic conservation laws," J, Comp. Phys., 49

(1983), pp. 357-393.

[7] A. Harten and S. Osher, "Uniformly high-order accurate non-osdllatory schemes, L," MRC
Technical Summary Report #2823. May 1985, submitted to SINUM.

[8] S. Osher, "Convergence of Generalized MUSCL Schemes," A''AS'A Langley Contractor Report
172306, (1984). SINUM v. 22, (1985), pp. 947-961.

[9] S. Osher, "Puemarm solvers, the entropy condition, and difference approximations," SINUM, v.
21, (1984), pp. 217-235.

[10] S. Osher and S. Chakravarthy, "Upwind schemes and boundary conditions with applications to
Euler equations in general geometries," J. Comp, Phys v. 50, (1983) pp. 447-481.

434

[11] S. Osher and S. Chakravarthy, "Very high order TVD schemes," ICASE Report #84-44, (1984),
Hampton, VA.

[12] S. Osher and S.R. Chakravarthy, "Hgh-resolution schemes and the entropy condition," SINUM,
V. 21, (1984), pp. 955-984.

[13] S. Osher and F. Solomon, "Upwind schemes for hyperbolic systems of conservation laws," Math.
Comp..v. 38 (1982), pp. 339-377.

[14] S. Osher and E. Tadmor, "On the convergence of difference approximations to conservation
laws," submitted to Math-Comp.

[15] P.L. Roe, "Approximate Riemann solvers, parameter vectors, and difference schemes," /. Comp.
Phys.. v. 43 (1981), pp. 357-372.

[16] P.L. Roe, "Some contributions to the modeling of discontinuous flows," in Lectures in Applied
Mathematics, v. 22, (1985) pp. 163-193.

[17] P.K. Sweby, "High resolution schemes using flux limiters for hyperbolic conservation laws,"
SINUM. v. 21, (1984), pp. 995-1011.

[18] E. Tadmor, "Numerical viscosity and the entropy condition for conservative difference schemes,"
NASA Contractor Report 172141, (1983), NASA Langley. Math Comp., v. 43, (1984).

[19] B. Van Leer, 'Towards the ultimate conservative difference scheme IV. A New approach to
numerical convection," J. Comp. Phys., 23 (1977), pp. 276-298.

[20] B. van Leer, 'Towards the ultimate conservative difference scheme V. A second order sequel to
Godunov's method," J. Comp. Phys., v. 32, (1979) pp. 101-136.

[21] B. van Leer, "Flux-vector splitting for the Euler equations," Proc. 8th International Conference on
Numerical Methods in Fluid Dynamics, Germany, June 28 - July 2, 1982.

[22] M. Ben-Artzi and J. Falcowitz, "An upwind second-order scheme for compressible duct flows,"
SiamJ. Sci. Comp., (1986) to appear.

435

[23] E, Harabetian, "A convergent series expansion for hyperbolic systems of conservation laws,"
NASA Contractor Report 172557, ICASE Report 485-13, 1985.

[24] S.R. Chakravarthy, A- Harten, and S. Osher, "Essentially non-oscillatory shock-capturing
schemes of uniformly very high accuracy," AIAA 86-0339, (1986), Reno, NA.

[25] A Harten, "On high-order accurate interpolation for non-osdllatory shock capturing schemes,"
MRC Technical Summary Report #2829, University of Wisconsin, (1985)

436

ON NUMERICAL DISPERSION BY UPWIND DIFFERENCING

Bram van Leer

Delft University of Technology

Delft, The Netherlands

ABSTRACT

Upwind-biased difference schemes for the linear one-dimensional convection
equation are defined. It is demonstrated that the numerical dispersion caused
by such schemes changes sign in the middle of the allowed CFL-number range.
This makes it possible to annihilate dispersive errors in two steps.

437

1. INTRODUCTION

Upwind differencing is a way of differencing convection terms. For
the scalar convection equation

u^ + au = 0, (1)

t X

discretized on a uniform grid {jAx,nAt}, the best-known upwind-differ-
ence approximation is the explicit first-order scheme of Courant,
Isaacson and Rees (CIR) [1],

n+1 n n n
u. - u. u. - u. .

' At ^ ^ ^ ' Ax' = °> a>0, (2.1)

n+1 n n n
u. - u. u. - u.

' At ' " ^ ' Ax ' = °' -<0- (2.2)

Introducing the Courant-Friedrichs-Lewy (CFL) number

a=a^, (3.1)

we may rewrite (2) as

where

u^' = (1 - |a|)u? + laju'? , (3.2)

J ' ' J ' ' J-s'

s = sgn a. (3.3)

438

The scheme is stable, even in the maximum norm, under the CFL condition

\o\ < 1. (4)

The value of u. given in (3.2) may be regarded as an approximation,
by linear interpolation, to the value of the exact solution

^T " "^"^j ~ ^^^' *^"^' ^^^

which, for non-integer 0, gets lost in the process of discretization.
The interpolation at t involves only the two nodal points nearest to
X. - aAx. Thus, the numerical domain of dependence of u, is upwind-
biased.

The upwind bias becomes more obvious as larger values of the CFL
number are allowed. If m is an integer such that

m < a < m + 1, (6,1)

a stable upwind scheme is [2]

• 1

u. = (m + 1 - a)uj_^ + (a - in)u^_^_j . (6.2)

Upwind differencing is often compared to central differencing,
where the numerical domain of dependence of u. at t" is centered on

439

X., the outcome usually being that upwind differencing is considered
superior but more complicated (because of the search implied in (6.1))
and central differencing inferior but simpler (no search needed) . Up-
wind differencing, it is said, stays closer to the physics contained in
the convection equation. If this indeed is desirable, one should be
able to measure the benefit. That, apparently, is not so easy: to
date, very few quantitative theorems have been proven supporting the up-
wind claim to a higher accuracy.

One piece of evidence can be found in [3] where Fromm's [4] "zero-
average phase-error" scheme (an upwind-biased scheme of second-order
accuracy) is shown to yield the lowest L^-error in convecting a step
function, in comparison to all other second-order schemes based on the
same data. This suggests the use of upwind schemes for shock-propaga-
tion problems, an area of application in which these schemes indeed are
unrivalled [5].

Another quantitative statement was presented by me without proof
in [6]; it concerns the lack of numerical dispersion by upwind schemes
at which Fromm already hinted. This will be the subject of the re-
mainder of the paper.

440

2. AN OPERATIONAL DEFINITION OF UPWINDING

To avoid cluttering up the formulas,! shall restrict the value of
the CFL number to the interval [0,1].

Definition. A scheme for Eq. (1) of the general form

is called upwind-biased for the CFL -number range [0,1] if its coeffi-
cients satisfy the symmetry relation

\(1 - ^) = c_j^-i(°)- (7.2)

Eq. (7.2) does not imply consistency of scheme (7.1) with Eq. (1);
for this we need to impose two more conditions:

I c, (a) = 1, (8.1)

k ^

I kc (a) = -a. (8.2)

k ^

A detailed analysis is needed to find the condition on the coefficients
that will ensure stability of the scheme for all values of a in the
range indicated.

441

It is possible to make scheme (7.1) yield the correct translated
initial-value distribution for integer values of 0; this clearly is
useful. The additional condition needed is

Cj^(O) = 0, k ^ 0. (9)

3. NUMERICAL DISPERSION BY UPWIND SCHEMES

When updating the solution with a scheme of the form (7.1), we
generally introduce both dispersive and dissipative errors. That is,
the Fourier components of the initial-value distribution are convected
by the scheme at the wrong speed, while also being damped. Only for
integer values of o these errors can be avoided simultaneously. For
non-integer values of a all consistent stable schemes of the form (7.1)
must be dissipative, since they are not invariant under time reversal.
With upwind-biased schemes at least the dispersion may be avoided, as
shown below.

Lemma. For any scheme that is upwind-biased for the CFL -number
range [0,1], the result of one step with CFL number a followed by a
step with CFL number 1 - a is free of dispersion.

Proof. Assume initial values according to

^ n laj , , -.

u = u^e -J; (10)

442

any upwind-biased scheme with CFL number a €. [0,1] may then be written

as

u"?"*"^ = g(a.a)u'? (11.1)

with amplification factor

K-1 . ,
g(a,a) = I cAOe^"^^, K> 1. (11.2)

k=-K ^

The same scheme applied with a CFL number 1 - O has an amplification
factor

K-1 . ,

g(l-a.a) = I c,(l-a)e^"''; (12.1)

k=-K ^

by virtue of (7.2) we have

K-1 . ,

g(l-a,a) = I c , ,(a)e^'"'. (12.2)

k=-K " '

Introducing H = -k-1 leads to

g(l-a,a)='f c,(a)e--(^^>)
£=-K ^

= e ^" I c„(a)e"

= e ^V(c^,a). (12.3)

443

The composite scheme, with a CFL number of 1, has an amplification
factor

-la

g(l-a,a)g(a,a) = e g*(a,a)g(a,a)

=e-i^|s(a,a)l2,

(13.1)

to be compared to the amplification factor for the exact solution at a
CFL number of 1 :

n+1 -ia n

u. = e u. .

J J

(13.2)

The two factors are identical in phase. Q

The above lemma has an interesting consequence.

Corollojy. An upwind-biased scheme for the CFL-number range [0,1]
has no dispersion for a CFL number of -y.

Proof. Apply the previous lemma to the case a = y. Since
a = 1 - a = -y, the two steps with the upwind scheme have the same
amplification factor

a-

g V.a = e

-ia/2

-,a)

(14)

with the correct phase -a/2, n

444

A geometric interpretation of this corollary for the CIR scheme is
given in Figure 1 .

(|j= -0 O

Figure 1. An illustration of the upwind property that arg g(a,a) =
-oa for \a\ = y, for the CFL scheme; the drawing is for
a = - — . The locus of g(a,a) is the circle (I) with
radius \o\ and center C in 1 - \o\ on the real axis;
arg g(a,a) is called ij;.

445

It further follows that for any value of a the dispersive error
changes sign when passes through (illustrated for the CIR scheme
by Figure 2), while the damping factor |g(a,a)| goes through an ex-
tremum [6], For all practical schemes this extremum is an absolute
minimum. Thus, in an upwind-biased scheme, minimum dispersion and
maximum dissipation go hand in hand. This, again, leads to the repre-
sentation of moving discontinuities with comparatively little ringing.

Besides upwind-biased schemes for a CFL-number range of the type
[m,m+l] there are upwind-biased schemes for the range [m-l,m+l]. These
are obtained by shifting the center of a central-difference scheme up-
wind over m meshes. An example is the fully one-sided, second-order
scheme for the CFL-number range [-2,0],

Uj"" = - I (1 - a)u^_2 + a(2 - a)u^_j + i (i - a) (2 - a)u'?. (15)

The coefficients of this scheme satisfy the relation

c^(2 - a) = c_^_2(a); (16)

Accordingly, a step with CFL number O should be followed by a step with
CFL number 2 - in order to achieve zero dispersion at a net CFL
number of 2.

For central-difference schemes the corresponding relation is

Cj^(-a) = c_^(a). (17)

hence, annihilation of phase errors cannot be combined with a net ad-
vancement in time.

UUf,

0.0

J LJ L

4 5

10

J L

15 20

50 £/Ax 100

Figure 2. Velocity dispersion versus wavelength for the CFL scheme.
The wavelength £ is related to a by a = 2ttAx/£; the
ratio of computed to exact convection speed is evaluated
as i|j/(-0a) .

447

REFERENCES

[1] R. Courant, E. Isaacson, and M. Rees, Comm. Pure Appl. Math . 5 (1952),
pp. 243-255.

[2] W. L. Miranker, Numer. Math . 17 (1971), pp. 124-142.

[3] P. Wesseling, J. Engrg. Math . 7 (1973), pp. 19-31.

[4] J. E. Fromm, J. Comput. Phys . 3 (1968), pp. 176-189.

[5] P. R. Woodward and P. Colella, J. Comput. Phys . 54 (1984), pp. 115-173.

[6] B. van Leer, J. Comput. Phys. 23 (1977), pp. 276-299.

448

AZTEG A FRONT TRACKING CODE BASED ON
GODUNOVS METHOD

BLAIR K. SWARTZ

and

BURTON WENDROFF

Theoretical Division, Group T-7, MS B284

Los Alamos National Laboratory

Los Alamos, NM 87545

ABSTRACT

AZTEC (Adaptive Zoom Tracking - Experimental Code) is a code to solve the
one-dimensional gas dynamic equations in a variable area duct with specific implemen-
tation for plane, cylindrical, and spherical geometries. The program uses a fixed, locally
and adaptively refinable grid, together with a set of moving grid points which migrate
through the fixed grid. The moving points represent shocks or contact discontinuities,
and they can be created or destroyed, usually as the result of a collision. Mass, energy,
and momentum (the iast only in the constant area case) are exactly conserved, except
after a collision; in that case the conservation error is reduced to invisible levels by spa-
tially localized partial time stepping. The basic difference scheme for both the fixed and
moving grid is Godunov's method, with the Riemann solver used to compute both cell
boundary fluxes and the speeds of the moving points. Tracking of rarefaction waves on
the moving grid is dif&cult with this method since the waves must be represented as
piecewise constant. In one version of AZTEC the rarefaction waves are recorded on the
fixed grid with the Lax-Wendroff difference scheme with a small additional viscosity,
and most of the numerical experiments have been performed with this version. In
another version the polytropic gas equation of state has been replaced by one in which
the pressure is a continuous piecewise linear function of specific volume at constant
entropy. With this assumption the solution of each Riemann problem is piecewise con-
stant, and our method is exact until the wave structure becomes too complicated. Some
preliminary numerical results are exhibited for this version.

• Sponsored by the U. S. Department of Energy under contract W-7405-ENG.36. The publisher recog-
nizes the U. S. Government retains a nonexclusive, royalty-free license to publish or reproduce the pub-
lished form of this contribution, or to allow others to do so, for U. S. Government purposes.

449

1. AFTER SOD.

Sod's survey paper [l] was a milestone in the development of numerical methods
for one dimensional gas dynamics, for it clearly exposed the shortcomings of some
methods which were in vogue at the time. It seems appropriate to point out that two
techniques which were not included in the survey are adaptive grid refinement and the
method of characteristics. Proper application of the latter requires some form of front
tracking, so that the programming of both methods is considerably more complicated
than for shock capturing schemes.

While just a modest amount of localized grid refinement will improve a shock cap-
turing method, there are pitfalls. We refer the reader to [2]. The method of characteris-
tics has two interpretations. In the first, the characteristic curves become coordinate
lines. Since there are three characteristics for the gas dynamic equations . two of them
must be chosen. The natural choices are the u+c and u-c characteristics. In the case of
isentropic flow, this means that differencing along the characteristics requires no inter-
polation. In the non isentropic case values on the third characteristic must be obtained
by interpolation.

The second expression of the method of characteristics is a form of upstream
differencing. The idea is roughly the following. Write the gas dynamic equations, or
any hyperbolic system, in the form w, + Aw^ = 0. Let Ij . j = l.- • ■ ji. be the left
eigenvectors of A , with eigenvalues X^ . Then

Ijiyv, +\^w^)= 0. (1.1)

This is differenced explicitly, using backward spatial differences for positive X^ and for-
ward differences for negative X; . More precisely,

Z;(wj"+i->Vj" + M;(>v<"-wt")) = (1.2)

where fij = \jAx/£u , and k=i—l if fij >0, k=i+l if fij <0. If there are discon-
tinuities present, they must be tracked through the grid in both versions.

AZTEC combines grid refinement and tracking, using conservative differencing.
The tracking is most easily done with Godunov's method, using moving grid points to
locate the discontinuities. A condition for the stability of Godunov's method is that the
fluxes on the cell boundaries remain constant during a time step. We found that the
simplest way to do this in our context was to remove fixed grid points near the moving
ones by locally coarsening the spatial grid. This is inaccurate if the moving point is in a
region with spatial variation, but we counteract that with a local grid refinement which,
450

as described later, refines in both space and lime. The Riemann solver, which provides
the fluxes for the conservative difference equations also determines the speeds of the
moving points, as suggested in [3].

In section 2 we give details of the grid refinement procedure. In section 3 we dis-
cuss the moving grid. In section 4 we exhibit the result of some compuutions. In sec-
tion 5 we present some preliminary results for a piecewise linear equation of slate.

2. GRID REFINEMENT.

The one dimensional gas dynamic equations for a variable area duct are

(a(x)p), +(a(x)pv), =0 (2.1)

(a (x )pv ), + (a (x )(pv2 + p )), = pa,

(a (x )pE ), + (a (x V ipE +/> )), = 0.

where p is the mass density, v is the velocity, E = e +(l/2)v2. e is internal energy, and
p is the pressure with equation of sute p = p (.p.e ). The quantity a (x ) is the area
function.

Our program was originally written for slab geometry. It was pointed out to us by
J.M.Hyman that an easy way to extend a fixed grid slab code to handle variable area is
to introduce area-weighted variables. Thus, we let

w = (a (x )p.a (x )pv ji (x )pE Y
so that the equations become

w,+(a/(w/a)), =g. (2.2)

where / is the flux vector given by

/ = (pv .pv2 + p ,v (p£ + /> )F

and

g = (O.pa, .OF

Suppose thai we have a uniform grid of N cells indexed by i. The quantity Wj"
will be the average of w in the cell at time n. Xj is the coordinate of the cell center;
x,+i/2 is the coordinate of the interface between cell i and i + 1. The area at a cell edge
is

451

°i +1/2 - O (x, +1/2).

but the area of the cell center is defined by

fli = a(Xi_i/2.Xi+i/2)

where

;(x,y) =

7

fais)ds (y -x)-i

and

a (x ^ ) = a (x ).

The basic conservative difference equation is

Ax wr +1 = Ax wr - Lt [{af ), +1/2 - (a/ )i -i/jJ+g Ax AT. (2.3)

If the cell interface with index i +1/2 is internal . that is, if cells t and » +1 are both
present on the grid, we allow two possible definitions of the numerical flux /j+1/2 •

The Godunov flux is obtained by solving the Riemann problem centered at x =
with left state given by Wj/oj and right state given by w,+i/a,+i. The flux function
evaluated at x = 0, r >0 is then used for /i+1/2.

The Lax —Friedrichs flux is defined as follows. Set

^i+m = -Siwi+i + Wi - Af /Ax(/j+i - /i )G(xi+i/2 )].
and then

/i+1/2 = / (w,- +1/2/0,- +1/2).

In the uniform area case if the fluxes at both cell boundaries are Lax-Friedrichs fluxes
then wC ■•"* becomes the two-step Lax-Wendroff' scheme.

The choice of fluxes is part of the experimentation with AZTEC. However, an
invariable strategy that we have implemented is to always use Godunov fluxes on the
finer grids (if they exist), at the external boundaries, and at cells in a neighborhood of a
moving grid point. If the Lax-Friedrichs flux is used at all on the coarse grid, it is in an
expansion region.

The grids are defined in terms of cells rather than points. The symbol j will
always identify a grid level, j = 1.2, • • • J . The maximum number of grid levels. / . is
an input parameter. Level 1 represents the coarsest grid, with A'(l) = N cells each of
length Ax (1) = Ax . Level 2 is a refinement of level 1 obtained by dividing each cell of

452

level 1 in half, so that Ax(2) = .5Ax(l), and A/(2) = 2A/(l). Thus.
Ax (;• ) = 2-<^ -i>Ax . and A^ (; ) = 2^ "'jV .

Since the refinement is local and adaptive, not all cells on every level will be
advanced at every time step. There are two kinds of cells, live and dead . At the start
of a time step the level 1 cells are all live. For ; >1. a cell on level ; will be live only
if its parent cell on level ; — 1 is live and if ceruin tests of the state variables on level
y-1 indicate that refinement (splitting) of the parent cell is required.

There will be two kinds of live cells, sterile and fertile . A sterile cell is one
which is not to be split and which therefore must be advanced by the difference equa-
tion. A fertile cell is one which splits into two daughter cells on the next level and
which is therefore not advanced by the differential equation. The advancement of a
sterile cell requires the computation of fluxes at the cell boundaries, but computation of
the flux at a fertile cell boundary will be needed only if that cell is not contiguous at
that boundary to a fertile cell on the same level. Since AZTEC is designed for serial
computation we have tried to avoid redundant calculation of fluxes. This arrangement,
which is not quite as complicated as it sounds, is shown schematically in Figure 2.1.

level

J

■I 1
1 1

< < 1
1 1

j +
j +

1

2

,

1
1

1

Fig. 2.1. Boundary fluxes at grid interfaces.

The boxes represent cells on the indicated grid level. The vertical sides of the boxes
represent the cell boundaries. A dotted line means that the flux is not computed on that
grid level. The flux is computed at a solid line. A solid line with an arrow means that
the flux computed at that grid is used at the next grid level. Thus, at an interface
between grids j and y+1 . the coarse grid flux supplies the boundary condition for the
finer grid. This generalization of [4] enables us to maintain conservation .

453

Since our difference scheme is explicit, the time step for level ; must be half that
for level j—1. An example of the evolution of the space-time grid is shown in Figure
2.2.

At

Ax
Fig. 2.2. Space-time refinement.

Note that refinement has occurred between the coarse time steps. Here is the algorithm
for setting up and advancing the grids.
BEGIN ALGORITHM

J-1

1 icCj) •• *ic(j) is for the first pass through level j, 1 for the second pass*

2 call CREATE(j) *Determine and label the fertile and sterile cells on level j, label
and provide data for the live cells on level j+1. Set nc(j+l) - number of live cells on
level j+I*

call FLUX(j) *Compute and store fluxes on level j at those interfaces which are a
boundary of at least one sterile cell.*

call ADVANCE(j) * Compute w" '•'^ on level j and overwrite on w" , for sterile cells
only*

if j < J and nc(j+l) pt

then j ■■ j+1 and go to 1

3 else if ic(j) -

then if j - 1

then step finished
else ic(j) - 1

454

go to 2
else j - j-1

call CONST(j) * The total conserved quantity (mass, for example) in a
fertile parent cell is defined to be the sum of the conserved quantities of
the two daughter cells.*
go to 3
END ALGORITHM

The subroutine CREATE requires further discussion. First, it must determine
which cells on level j are to be split. This is done by performing two tests for each cell.
If I is the cell index, then one of the tests looks for moving grid points in the cells
Z -2. Z -1, i . / +1. Z +2 (see section 3). If there are any. then cell Z splits into two. Of
course, special provisions have to be made for cells close to the boundaries. The second
test splits cell I if there is a compression in the same neighborhood as above; other cri-
teria could be included. Now. suppose that in advancing from t to f + Af CREATE
finds that a cell on level j must be split into two daughter cells on level y +1. There are
two possibilities: the daughter cells were present at the previous time step and were
advanced to time t by the algorithm, or they were not. In the former case no new data
need be created for the daughters. In the latter case data is obtained by interpolation. If
the parent cell on level j has index i . the interpolation is as follows. Let L and R be
the indices of the left and right daughter cells, respectively, and let

Wl = 1.25wj - .25(aj/ai+i)w,+i.

wjj = .75wj+.25(aj/aj+i)>Vi+i

If both Wi/fli and wg/aji lie between Wj_i/aj_i and Wi+i/aj+i. accept w^ and Wg as
the interpolated values. If not, let

wi = .75w,- + .25(aj/aj_i)w,_i

vtg = 1.25>Vj — .25(aj/ai_i)wi_i.

Use these as the interpolated values unless the above monotonicity test fails, in which
case set

vfi = iai/aiywi^

Wg = (Cjf /Oj Vi .

The latter is also xised if cell i is at the boundary of the physical domain.

455

3. THE MOVING GRID.

The moving grid points will move through the fixed grid and exchange data with
the fixed cells. We have chosen to do this for the finest grid only: this is arranged by
having one of the refinement tests look for moving points in the two cells on each side
of the current cell. If the points are not allowed to move more than the length of one
cell in one time step, they cannot leave the fine grid.

The moving gridpoints define boundaries of skewed space time cells in which the
conservation laws are applied just as they were for the fixed grid. For the two points
X < y shown in Figure 3.1, the difference equation is

[y -X H(Ty -o-;, )tu ]w^y = (y -X )w^y - At [a (y .y +cry At )Fj -a(xjc +cr^ At )F„ ]

+ |"Ar. (3.1)

X

►-

1

-^ —

-•■xy —

~"*"

\

Speed
°-x-

/

- *xy

\

^ Speed
ay

/

1 *

\

X y

Fig. 3.1. Space-time cell defined by moving points.

The quantity w^j is both the right value for the discontinuity at x and the left value
for the one at y. The term gAt only appears in the momentum equation. To avoid
false accelerations it must have the following form. If, in the momentum equation,

and similarly for F^ . then

g = (l/2)(;», + py )[a (y ,y + o-y Ar ) - a (x ,x + 0-, At )].

There are two things that must be provided in the basic difference equation above:
the speeds ct and the fluxes F . These are obtained from the solution of Riemann prob-
lems. In order to do this we must first recover the hydrodynamic variables from the
area weighted variables. Suppose, for the moment, that we have done this properly.
Then when the grid point at position x is to be moved there will be associated with it a
left state u_ and a right state u+. We find the complete solution of the Riemann

456

problem for these two slates. Then we decide which ray or rays are to be followed. For
example, if the point x is a contact discontinuity and if the solution of the Riemann
problem has a sufficiently strong contact discontinuity, then we take the new speed to
be the speed of the contact. The new flux F is f — aru evaluated on this ray (this takes
account of the fact that the ray is not necessarily vertical in the space-time plane). Note
that / — cru is continuous across every ray in the solution of the Riemann problem.
More generally, the point x might spawn several new moving points. If x is the result
of a collision with another point or with a reflecting boundary, then we could follow all
the shocks and contacts which emerge.

The complete logic of the procedure for deciding which rays to keep is too compli-
cated to give in complete detail here, but we can give an outline of it. First, the Riemann
solver produces a list of speeds and fluxes and identifiers for each sufficiently strong
wave which is present in the solution. Thus, a shock corresponding to the characteristic
V +c is identified as a 3-shock. and a speed and flux are given for it. A rarefaction
corresponding to the characteristic v — c is identified as a 1-wave, and for it the speeds
and fluxes on the leading and trailing edges are provided. Next, tactical decisions are
made in a subroutine called TRACK, which has the job of creating and destroying mov-
ing points, advancing the moving points and checking for collisions, maintaining stabil-
ity on the moving grid, and communicating with the most refined portions of the fixed
grid.

Here is how TRACK works. First, the points are collected into blocks. Each block
is such that the rightmost point of one block is separated by five or more full fixed cells
from the leftmost point of the next block, as in Figure 3.2.

Fixed cell
boundaries

L J O I O I 1 1 \ I I p n I

^ Moving points ^Moving points

in one block in next block

Fig. 3.2. Blocks of moving points.

Each block is processed independently of the others, so what follows refers to the
points of one block. In order to improve resolution in the variable area case, if two

457

adjacent points are two or more full cells apart some fixed grid points between the pair
are treated as moving. These are called separator waves. Next, each moving point is pro-
vided with a left and right average value of the area-weighted variable w so that if
w_(x ) and w+(x ) are respectively the left and right sutes of x . and if x and y are
adjacent points (x < y ) then w+Cx ) = w.Cy ). This is done using a combination of fixed
cell data and moving point data obtained from eq. (3.1) for the previous time step,
depending on the separation of the points. Of course, this is done conservatively. The
hydrodynamic variable corresponding to w+U ) is u+(x ) := w Jix )/a (x .y ). Now the
Riemann solver is called for each point in the block. For the typical grid point all the
rays returned by the solver are assumed to define new points which are inserted into
the list of moving points. There are exceptions to this; for example, at a left reflecting
boundary only rays with non-negative speeds are retained. The list is ordered by posi-
tion if the positions are unequal and by speed otherwise, as in Figure 3.3.

Fig. 3.3. Ordering of moving points.

At this stage we have many more points than we want or need, but most of them will
be deleted at the end of the time step.

The reason for retaining so much information is that this gives us a procedure for
maintaining stability during a collision or close approach of moving points. If a collision
occurs at time « + Sf . < St < Af , the current block of points is advanced to f +Bt
using eq. (3.1) with A/ replaced by 8t . Then we attempt to finish the time step by
advancing from t +St to t + Ar , checking again for a collision, etc. The use of blocks
causes this partial time-stepping to be spatially localized, unless the moving grid is
evenly distributed in the fixed grid. The idea now is that any collision which occurs at
this time is "exact." which means the following. If the points x and y in Figure 3.1

458

collide at lime t + ^t . then x + cr, Af =y +a-yAt . so that the left side of eq.(3.l) is
zero. On the other hand, even if the source term were not present, the right side of that
equation cannot be expected to be zero. Indeed, consider the case shown in Figure 3.4.

Speed s

I -shock

Fig. 3.4. Exact collision.

In this situation, a contact and a 1-shock have arrived at x and y . respectively. The
Riemann solver has produced at x a contact with speed s and a 3-rarefaction wave,
while at y the solution is a 1-shock and some other waves that play no role. Two bad
things happen if we suppress the rarefaction wave. First, because the solution is not
constant along the ray yz we can expect an instability to develop. Second, making the
appropriate substitutions into the right side of eq.(3.l). we have

i? 1 := (y - X )a (x ,y )uo - Sr [a (x ,2 )5u 1 - a (y ,2 )o-, UqI

St [a (y j)fiuo)-a{xj )/ (u j)].

Even if the area factors were constant, this would not be zero unless Ui = Uo- On the
other hand, if we include the leading edge of the rarefaction in the list of moving
points, then the first collision occurs at c. Then the right side of eq.(3.l) becomes

R 2 := Stf (u o)[a Cy .y +o-y St)-aixjc +0-, St )].

In the constant area case. i?2 = 0, hence the nomenclature exact. In other words, an
exact collision is one in which the state between the two intersecting rays is constant.
For such a collision eq.(3.l) is identically correct if the area is constant. When the area
is variable, the error in the mass and energy conservation is second order in the mesh
size.

459

In Figure 3.5 we can see how the collision between the contact and the shock will
actually occur.

y

Fig, 3.5. Collision and precursors.

After several partial time steps, caused by collision of the precursor rarefaction wave
with the shock, the rarefaction wave will become too weak to be seen by the Riemann
solver and the main collision will take place. There will be a small error in a conserved
variable such as the mass. The program controls this error by two devices. If the error
exceeds a pre-set value the time step is repeated with a smaller strength threshold in the
Riemann solver. This works well for constant area, but is not enough in other
geometries. For them, we must force additional partial time steps that will reduce /?2-

At the end of the time step (partial or complete) the precursor waves are deleted.
At the end of a full time step the separator waves are also removed. Thus each step
starts fresh with the main moving points. However, if a major collision has occurred,
points may have been created or destroyed. If we wish to keep track of all shocks and
contacts, then we must include the resulting transmitted and reflected shocks and resi-
dual contact produced by a collision of two shocks or of a shock and a contact. The
entire process that we have described works remarkably well, particularly if collisions
are rare.

4. THE TEST PROBLEMS.

Three test problems are presented. The first is Sod's problem with reflecting boun-
daries [l]. The second is a problem posed by Paul Woodward [5] involving the interac-
tion of the solution of two Riemann problems. The third is an elegant spherical shock

460

problem with a simple exact solution due to Bill Noh [6].

In Figure 4.1 we give our solution (density only) of Sod's problem at t -.175. The
initial data define a Riemann problem centered at x - .5. The left state has density 1.0.
pressure 1.0, and velocity 0.0. The right state has density .125, pressure .1. and velocity
0.0. The equation of state is that for a y - law gas with -y - 1.4. This initial-value prob-
lem resolves into a rarefaction wave, a contact discontinuity, and a shock wave (from
left to right).

1.0-

0.8 ■

0.6

0.4

0.2

0.0

0.0

0.2

0.4

0.6

0.8

1.0

Fig. 4.1 Sod's problem at f = .175 .

The shock is correct, but the state between the contact and the rarefaction is in
error by 5%. This is caused by the presence of the strong rarefaction in close proximity
to the contact early in the calculation.

In Figure 4.2 we give the apparently converged computed solution at t - .81. By
this time the main shock has reflected off the right boundary and interacted with the
contact, producing reflected and transmitted shocks. The rarefaction has reflected off

461

the left boundary, begun to emerge from the interaction with its image, and is just now
beginning to interact with the main pair of reflected/transmitted shocks.

i.o-

0.8-

0.6-

0.4-

0.2-

0.0

0.2

O.A

0.6

0.8

1.0

Fig. 4.2 Sod's problem at r = .81 .

The initial conditions for Woodward's problem are a gas at rest with unit density
in a unit interval with reflecting walls. The pressure in the left-most 1/10-th of the
interval is 1000 and the pressure in the right-most 1/10-th is 100: it is .01 otherwise.
The initial rarefaction waves moving toward the boundaries reflect and quickly catch
up to the contacts and the shocks. The collision of the shocks and their trailing waves
at about t = .028 initiates a complex sequence of intense interactions localized within
five to twenty percent of the interval. The computed density is shown in Figure 4.3 at
t = .038 . Woodward has computed this with a very fine grid, but he only gives a
graph of the solution. We differ from his solution only in the magnitude of the peak
density, which he finds to be 6.5 while ours is 7. Both calculations locate the

462

discontinuities in the same places.

8

0.0

0.2

^ \ ' \ ^

0.4 0.6 0.8 1.0

Fig. 4.3. Woodward's problem at t = .038 .

For Noh's problem we have a sphere of unit radixis filled with a y-law gas. y -
5/3. at zero pressure and internal energy, and with velocity - -1. At t = .6 the solution
consists of a shock located at z » .2 moving with speed 1/3. Behind the shock the pres-
sure is 64/3 and the density is 64. Ahead of the shock the density \sl+t /r^. The com-
puted density is given in Figure 4.4.

100

Density

-I I — I — r— i — I — I — I — r

0.0 0.5 1.0

Fig. 4.4. Noh's problem.

463

5. THE PIECEWISE LINEAR EQUATION OF STATE.

The approximation of arbitrary functions by piecewise linear ones has a long and
distinguished history. The value of this approximation in the theory of conservation
laws seems to have been first recognized by Dafermos [7], who combined a piecewise
linear flux function and piecewise constant initial data to obtain an elegant existence
theorem for scalar conservation laws. The crucial property of the piecewise linear scalar
flux is that the solution of the Riemann problem has only conicant states. Hedstrom [8]
observed that if the pressure expressed as a function of specific volume and entropy is
piecewise linear in the volume, then again the solution of the Riemann problem has only
constant states. Hedstrom used this as a computational device to obtain numerical solu-
tions of the equations of isentropic flow, by tracking the shock-like boundaries of the
constant states. In principle, AZTEC can obtain the exact solution of the full gas
dynamic equations with such a piecewise linear pressure and piecewise constant initial
data simply by having no fixed grid points, only moving ones. If we also take a very
large lime step, then the collisions determine the intermediate time steps. Each collision
will be exact in the sense defined in section 3.

In Figure 5.1 we show the solution of Sod's problem for a piecewise linear approx-
imation to the y - law gas.

1.0

0.8

0.5

0.4

0»2

0.0

0.000 0.125 0.250 0.375 0.500 0.625 0.750 0.875 1.000

Fig. 5.1. Sod's problem for a piecewise linear equation of slate, t = .175

464

There are 80 nodes per decade in density. The shock and contact are now exact (for the
given equation of state), and the rarefaction wave has become piecewise constant. Of
course, this is a trivial application of the method as there have been no collisions.

In Figure 5.2 we have a more interesting example, namely. Sod's problem with
reflecting boundaries at t=.81. computed with 80 nodes per decade in density. Now we
have a rarefaction wave reflected off the left boundary and interacting with the waves
reflected from the other boundary. This result should be compared with Fig. 4.2. The
solution in Fig. 5.2 was obuined about 30 times faster than the one in Fig. 4.2.

1.0

0.8

0.6

0.4

0.2

0.000 0.125 0.250 0.375 0.500 0.625 0.750 0.875 1.000

Fig. 5.2. Sod's problem, f = .81 . piecewise linear eqtiation of state.

There are serious difficulties with the piecewise linear method which seem to
prevent it from being more than a curiosity. One problem is that, in general, a collision
of two waves will prodjice at least three outgoing waves, leading to a rapid prolifera-
tion of waves and collisions. This can even happen with the interaction of rarefaction
waves, such as occurs in the reflecting Sod problem. In the case of the interaction of
rarefaction waves, this difiSculty is overcome by the following device. Suppose the pres-
sure p at fixed entropy is continuous, and that it is linear in the intervals (t.-.Tj+j),
i = 1,2 • • • ,n , where T = 1/p. We constrain the nodal values />,• to satisfy the condi-
tion that ipi — i>i+i)(Ti+i — T, ) is a constant that depends only on the entropy, not » . It

465

follows from this that the velocity jump across each internal ray in a rarefaction fan is
constant. Moreover, an examination (as in [8]) of the rarefaction curves in the
pressure-velocity space shows that the number of collisions in the complete interaction
of two rarefaction fans is now of order n^ if there are n nodal pressure values in the
fans. This constraint was used to generate Figures 5.1 and 5.2.

We anticipate reporting additional detail on our experie-^ces with piecewise linear
equations of state.

REFERENCES.

[l] G. Sod, "A survey of several finite diflFerence methods for systems of

nonlinear hyperbolic conservation laws." J. Comp. Phys. . Vol. 27 (1978). pp. 1-31.
[2] B.K. Swartz. "Courant-like conditions limit reasonable mesh refinement

to order h^ ." Los Alamos National Laboratory Preprint LA-UR-8 1-2037. 1981.
[3] S.K. Godunov. A.V. Zabrodin. and G.P. Prokopov. "A computational

scheme for two-dimensional non stationary problems of gas dynamics

and calculation of the flow from a shock wave approaching a stationary state."

USSR Comp. Math. Math. Phys. . Vol. 1 (1962). pp. 1187-1219.
[4] G. Browning. H.-O. Kreiss. and J. Oliger. "Mesh refinement." Math. Comp. .

Vol. 27 (1973). pp. 29-39.
[5] P.R. Woodward, "Trade-ofiFs in designing explicit hydrodynamic schemes

for vector computers." Livermore National Laboratory Preprint UCRL-85813. 1981.
[6] W.F. Noh. Artificial viscosity (Q) and artificial heat flux (H) errors for

spherically divergent shocks. Lawrence Livermore National Laboratory

Preprint UCRL-89623. 1983.
[7] CM. Dafermos, "Polygonal approximations of solutions of the initial-value problem

for a conservation law." J. Math. Anal. Appl. . Vol. 38 (1972). pp. 640-658'.
[8] G.W. Hedstrom. "Some numerical experiments with Dafermos's method for

nonlinear hyperbolic equations," Lecture Notes in Math. . Vol. 267 (1972). Springer.

Berlin.

466

LEAST SQUARES FINITE ELEMENT SIMULATION
OF TRANSONIC FLOWS

T. F. Chen
Carnegie-Mellon University

G. J. Fix
Carnegie-Mellon University

ABSTRACT

Finite difference approximation of transonic flow problems is a well-
developed and largely successful approach. Nevertheless, there is still a
real need to develop finite element methods for applications arising from
fluid-structure interactions and problems with complicated boundaries. In
this paper we introduce a least squares based finite element scheme. It is
shown that, if suitably formulated, such an approach can lead to physically
meaningful results. Bottlenecks that arise from such schemes are also
discussed.

Research was supported in part by the National Aeronautics and Space
Administration under NASA Contract Nos. NASl-17070 and NASl-18107 while the
second author was in residence at the Institute for Computer Applications in
Science and Engineering, NASA Lanley Research Center, Hampton, VA 23665-5225.
Partial support was also provided by Army Research Office under Contract No.
DAAG29-83-K0084.

467

1. INTRODUCTION

In this paper we consider the approximation of transonic flows by finite
element methods based on a variational method of the least squares type. The
objective here is purely computational. In particular, we have sought to
fully exploit the ideas arising from mathematical analysis of such methods
(see, for example, [1] - [6]) and directly apply them to a nontrivial
transonic flow problem. The major conclusion drawn from this work is that
finite element methods — suitably formulated — can give physically meaningful
results.

There is a significant and largely successful array of finite difference
techniques for transonic flows (e.g., [17]). Nevertheless, an assumption
implicit in this work is that there is still a need for stable and accurate
finite element approaches. First, there are applications from fluid-structure
interactions that would benefit from the availability of a finite element flow
model. Second, there is the issue of complicated boundaries in the flow
field. The importance of the finite element ideas in such a context — while
largely untested — is still promising.

Variational principles of the least squares types have a number of
valuable computational properties. For example, the algebraic system
generated is always Herraitian semidef inite. In addition, such schemes, if
properly formulated, are insensitive to equation type, be it hyperbolic
(supersonic flows) or elliptic (subsonic flows). In fact, the majority of the
finite element ideas that have been used for hyperbolic problems to date tend
to be either implicitly or explicitly of the least squares type.

Least squares based schemes do have, however, some major computational
defects. First, they tend to be sensitive to singularities and

468

discontinuities in the flow variables. Moreover, mesh refinement alone does
not overcome these defects [7]. Based on the work in [7] we introduce
weighted least squares variational principles, which in combination with mesh
refinement is capable of dealing with shocks in the flow field.

In Section 2 we describe the basic numerical formulation, and outline the
essential computational properties associated with the approach. A key
feature is the proper choice of weighting functions to use in the least
squares functional. A closely allied issue is the density modifications
needed to rule out nonphysical expansion shocks.

In Section 3 we present sample numerical results. As a model problem we
select the planar potential flow over a cylinder.

Other authors have considered finite element approximation of transonic
flows. Selected references are [18] - [21].

2. THE LEAST SQUARES FORMULATION

We consider the potential flow over a body fi. Let u^ denote the
velocity and p the density. Then a mass balance yields

div[pu] = 0. (2.1)

In addition, we have

u = grad ({> (2.2)

for the velocity potential <(). The density p is given as a function of u
by the Bernoulli equation. The system is closed by specifying the normal
velocity

A69

u«n = V

(2.3)

at the boundaries of the flow region. On the body ^ the no flow condition

ii'ii =

applies. We assume that the flow region is contained in a box B and that
(2.3) is specified on the boundary of B. Thus

A

fi = B/n (2.4)

defines the flow region, and (2.1) - (2.2) hold in Q with (2.3) holding on
the boundary T and ^.

Since the flow is assumed to be irrotational, (2.1) - (2.2) can be
replaced with

div(pu) = in n (2.5)

curl(u) = in n (2.6)

u«n = V on r. (2.7)

A least squares scheme based on this system takes the form

/ {|div(pu)|^ + |curl(u)|^} = min, (2.8)

where the variation is taken u in some finite element space satisfying the

470

boundary conditions (2.7). Such a div - curl system has proven to be very
effective for elliptic systems (subsonic flows) in cases where the density
P = p(u) and the velocity field _u are smooth [8].

Preliminary results indicate that with appropriate weighting functions on
the terms in (2.8), the nonsmooth cases can be treated as well. Nevertheless,
In this paper we shall focus attention on (2.1) - (2.2) and least squares
schemes of the form

/

V

— - grad (|)

2 21

+ wjdiv v| V = min, (2.9)

where v = pu is the mass flow and w is a weighting function to be chosen.
In this setup the variables are the potential <j) and the mass flow v.
The density in (2.9)
'. . P = p(|grad <|)|)

is obtained from Bernoulli's equation, i.e..

P^-^ =

[ -{^)^l (Igrad*|2- 1)1

Thus, (2.9) is a nonlinear least squares formulation, which is appropriate
since it reflects the nonlinear character of transonic flow. Once a grid is
selected (specific examples are given in the next section), the minimization
of (2.9) over the associated finite element space leads to a nonlinear system

K($^)$^ = F. (2.10)

471

In all of the numerical examples reported in the next section, (2.10) was
solved by a combination of Newton's method and elimination. Issues related to
this choice for the equation solver will be discussed in the next section.
There are three main cases that are considered in this paper:

Case 1 ; smooth subsonic flows,

Case 2 ; smooth transonic flows ,

Case 3 ; transonic flows with shocks.
In the first case (2.9) can be used without modification, and in
particular no weighting function is needed (i.e., w e 1 can be used). One
does need special grids to obtain optimal accuracy (see [1]), and the criss-
cross grid pattern which satisfies the grid decomposition property of [1] is
used.

In the second case a hyperbolic region appears but the flow field remains
smooth. In this case there is a loss of accuracy in the hyperbolic region.
In particular, with linear elements the pointwise accuracy in the mass flow v^
drops from O(h^) — in a generic mesh spacing — to 0(h). This can be corrected
with a suitable choice of weighting function w, and details are given in [8].
This modification was not used in the results reported in this paper since the
hyperbolic regions in question were too small for the suboptimal accuracy to
have a major effect on the qualitative features of the flow.

The third case is, by a wide margin, the most important as well as the
most challenging. Here we, use a weight w so that the term

/ < w|div _u| +

u

- grad <{)

(2.11)

472

remains meaningful. In addition, modification to the density p = p(|grad (j) | )
must be introduced so that nonphysical expansion shocks are eliminated.

For the choice of the weight w, we follow the developments introduced in
[7]. For most flows, v = pii is continuous across the shock [10]. Neverthe-
less, it does not follow that div v^ is square integrable, and the primary
rule derived from [7] is that w be chosen so that

/ w|div v|^ < ». (2.12)

This requires that w vanishes appropriately on the shock, which in turn
means that (2.11) is a least squares principle in a degenerate L norm. A
point of significance, on the other hand, is the fact that if w vanishes to
minimal order on the shock (in that (2.12) still holds), then optimal O(h^)
can be achieved in unweighted L norms provided appropriate mesh refinement
is introduced. This has been proved rigorously only in special cases (see
[7]), yet the numerical results in the next section seem to indicate that the
principle is general.

These modifications alone do not yield an accurate simulation of the flow
problem. To do this one must deal with the presence of nonphysical expansion
shocks. In effect, (2.9) does not have a unique minimum, neither over
infinite-dimensional function spaces nor over the finite-dimensional finite-
element spaces. One can have expansion shocks, compression shocks, or both.
What is interesting is the results In the next section tend to indicate that
the case where both type of shocks appear tends to be the stable mode for
(2.10). That is, an arbitrary choice of starting vector for Newton's methods
applied to (2.10) tends to converge to this solution.

473

To eliminate expansion shocks we consider density biasing which in effect
introduces streamwise diffusion into (2.1) - (2.2). Following [11] (see also
[12] - [14]) the modified density takes the form

p = p - MP As, (2.13)

where p is the derivative of the density p along the streamwise
s

direction. Since the density has the form

p = p(|grad (t)|),

the derivative p formally involves second derivatives of (|). Since <t) is
expanded in terms of linear elements, it is necessary to replace p with a
streamwise difference quotient; i.e..

p = p - U Ap As, (2.13')

in the least squares formulation.

3. NUMERICAL RESULTS

To illustrate the above ideas we selected the classic problem of a planar
flow past a cylinder. The flow region plus boundary conditions are given in
Figure 3.1. The configuration shown in this figure assumes that both the
outflow and inflow remain subsonic. Figure 3.2 contains a typical grid. For
economy only the top part of the flow region is shown, and the special
refinement needed for the shocks is not shown.

474

The first set of results shows a typical subsonic flow pattern. The
results are given in Figure 3.3 for a free stream Mach number of

M = 0.1.

00

Convergence studies at such Mach numbers are reported in [5] - [6]. These

results indicate, with the type of grid shown in Figure 3.2, one can readily

2
achieve L error of 1% or less for the velocity field.

The next set of results deal with the smooth transonic case. Of special
interest here is the ability of the scheme to detect the onset of supersonic
flow. Analytical techniques (see [15] and [16]) have given accurate values
for the critical free stream Mach number M* as a function of d/D, where
d is the diameter of the cylinder and D is the width of the channel. These
results are reproduced in Figure 3.4. Numerical results from the least
squares scheme are given in Figures 3.5 - 3.7 for M^ = .42, .45, and .50,
respectively. The d/D ratio used for this case is 1/6. Extrapolation
based on these results indicates that the critical Mach number is
approximately .41, which is good agreement with Figure 3.4.

The next set of results show what least squares based schemes produce when
diffusion via density modification is not used. These are shown in Figure 3.8
which contains plots of the velocity q = |u| versus angle Q along the
cylinder and at a radius slightly above the cylinder. The free stream Mach
number is M^ = .5. The shock at the front of the cylinder is an expansion
shock and is nonphysical. The one at the rear is a compression shock. A
remarkable feature of this approximation is that the physically relevant
compression shock is approximately in its correct position and is apparently
unaffected by. the spurious shock. (Compare Figures 3.8 and 3.9.)

475

The solution shown in Figure 3.8 is apparently a stable mode for the
nonlinear system (2.10). Indeed, Newton's method converged to this solution
rather rapidly for a wide variety of initial conditions.

In this regard, it is interesting to note that for the least squares
formulation the Jacobian is not singular near the solution shown in Figure
3.8. Density modifications are needed to remove the spurious shock shown at
the front of the cylinder. However, they are not needed to obtain nonsingular
Jacobians.

The final results deal with the complete least squares system with the
density modification discussed in the previous section. Figures 3.9 - 3.11
show the velocity field over the cylinder, at a radius slightly larger that
that of the cylinder, and at a radius in the free stream. Note that the
spurious expansion shock has been totally eliminated. Moreover, the shock
location and strength as well as the velocity profile appear to be correct as
is the supersonic bubble shown in Figure 3.12.

While we regard these numerical experiments as successful, there are a
number of areas where the approach could be improved. The first issue
concerns the equation solver. Once the density modification were introduced,
the number of iterations increased by a factor of 2 to 3. Moreover, the
solution shown in Figure 3.9 tended to be less "attractive" to the Newton
iterations than that shown in Figure 3.8 (without density modifications). In
fact, it was not difficult to find starting vectors where nonconvergence was
seen, in the former case, although the starting state of a uniform flow always
leads to convergence. This suggests that an alternative equation solver
(e.g., preconditioned conjugate gradient) might be a more efficient choice for
the equation solver.

476

A second issue concerns post-shock oscillations. These are seen in Figure
3.10, which is the radius where the oscillations were found to be the most
significant. These oscillations were not seen on the body of the cylinder
(Figure 3.9) and disappeared rather rapidly away from the cylinder (Figure
3.11). This is clearly a grid effect due to the slight misalignment of shock
and grid.

4. CONCLUSIONS

Finite difference approximations to transonic flow problems are well-
developed and have been successfully used for a wide range of problems.
Nevertheless, there is still a need to develop finite element approaches for
such problems for a variety of applications. We feel that the results
presented here do show that such schemes can give physically meaningful
simulations.

On the other hand, our experience has tended to indicate that
straightforward application of the basic finite element idea may not always be
successful. Key computational issues are as follows:

(i) There is a need to carefully develop the spaces in which the
approximations are formulated. Classical L spaces are generally
inappropriate,
(ii) Some form of diffusion (via density modifications or otherwise) appears
to be needed. Moreover, care is needed in the way this diffusion is
introduced,
(iii) The geometrical pattern of the grid selected is of importance. Some

477

patterns are definitely superior to others.

Finally, there are some important "bottlenecks" associated with the scheme
employed in this paper, which, if properly addressed, could lead to an even
more efficient approach. These include the following:

(i) There is a need for an equation solver that is more efficient than the
Newton method used in this paper,
(ii) There is a need for adaptive grid refinement techniques that would lead
to a better shock grid alignment than that achieved in this paper.

478

REFERENCES

[1] G. J. Fix, M. D. Gunzburger, and R. A. Ntcolaldes: Least squares finite
element methods, NASA-ICASE Report No. 77-18, revised version published
in Comput. Math. Appl. , Vol. 5, 1979, pp. 87-98.

[2] G. J. Fix and M. Gurtin: On patched variational methods, Numer. Math .,
Vol. 28, 1977, pp. 259-271.

[3] G. J. Fix and M. D. Gunzburger: On least squares approximation to
indefinite problems of the mixed types, Internat. J. Numer. Methods
Engrg ., Vol. 12, 1978, pp. 453-470.

[4] C. L. Cox, G. J. Fix, and M. D. Gunzburger: A least squares finite
element scheme for transonic flow around harmonically oscillating wings,
J. Comp. Phys ., Vol. 51, No. 3, September 1983, pp. 387-403.

[5] T. -F.Chen: On finite element approximations to compressible flow
problems, Ph.D. Thesis, Carnegie-Mellon University, May 1984.

[6] T. F. Chen: Least squares approximation to compressible flow problems,
submitted to Comput. Math. Appl .

[7] C. L. Cox and G. J. Fix: On the accuracy of least squares methods in the
presence of corner singularities, Comput. Math. Appls ., Vol. 10, No. 6,
1984, pp. 463-476.

479

[8] G. J. Fix and M. E. Rose: A comparative study of finite element and
finite difference methods for Cauchy-Riemann type equations, SIAM J.
Numer. Anal .. Vol. 22, No. 2, 1985, pp. 250-260.

[9] G. J. Fix: Least squares approximation o hyperbolic systems, submitted
to SIAM J. Numer. Anal .

[10] P. D. Lax: Hyperbolic Systems of Conservation Laws and the Mathematical
Theory of Shock Waves , SIAM Regional Conf. Series Lectures in Appl.
Math., Vol. 11, 1972.

[11] S. Osher, M. Hafez, and W. Whitlow, Jr.: Entropy conditions satisfying
approximations for the full potential equation of transonic flow. Math.
Comp. , Vol. 44, No. 169, January 1985, pp. 1-29.

[12] A. Eberle: Eine Method Finlter Elements Berechnung der Ttanssonicken
Potential— Strimung un Profile , MBB Berech Nr. UFE 1352(0), 1977.

[13] M. M. Hafez, E. M. Murman, and J. C. South: Artificial compressibility
methods for numerical solution of transonic full potential equation,
AIAA Paper 78-1148, Seattle, Washington, 1978.

[14] M. Hafez, W. Whitlow, Jr., and S. Osher: Improved finite difference
schemes for transonic potential calculations, AIAA Paper 84-0092, Reno,
Nevada, 1984.

480

[15] I. Imai: On the flow of a compressible fluid past a circular cylder,
II, Proc. Phys. Math. Soc. Japan , Vol. 23, 1941, pp. 180-193.

[16] Z. Hasimoto: On the subsonic flow of a compressible fluid past a
circular cylinder between two parallel walls, Proc. Phys. Math. Soc.
Japan , Vol. 25, 19A3, pp. 563-574.

[17] A. Jameson: Numerical solutions of nonlinear partial differential
equations of mixed type. Numerical Solutions of Partial Differential
Equations III , Academic Press, New York, 1976, pp. 275-320.

[18] M. 0. Bristeau, R. Glowlnskl, Periaux, J., P. Perrier, 0. Plronneau,
G. Poirier: A Finite Element Method for the Numerical Simulation of
Transonic Potential Flows, Finite Element Handbook , McGraw-Hill, 1983.

[19] R. Pelz and A. Jameson: Transonic flow calculations using triangular
finite elements, AIAA J ., Vol. 23, No. 4, 1985, pp. 569-576.

[20] W. G. Habashi and M. M. Hafez: Finite element solution of transonic
flow problems, AIAA Paper 81-1472,

[21] H. Deconinck and C. Hirsch: Finite element methods for transonic flow
calculations, Proc. Conference on Numerical Methods in Fluid Mechanics ,
3rd, Cologne, West Germany, October 10-12, 1979, Braunschweig, Friedr.
Vieweg und Sohn, Verlagsgesellschaf t mbH, 1980, pp. 66-77.

481

u^=v

u^=v

U2=0

Figure 3.1. The flow region fj.

482

\

\

Ay

\ /A

/

M

/

M

.^^

^M

\ly/\ / ^^<^

S^V/^^^/

^ X \.y\iy\ly

W

^^

Figure 3.2. 512 elements, 281 nodes, h = 0.30907 x 10

-1

A83

6.00

4.00

2.00

2.00

J L

4.00

10. 00 12. 00

Figure 3.3. Flow pattern for the free stream Mach number M =0.1.

484

0.6

M

#

0.5

0.4

0.3 -

0.2

%

"^

\

0.1 0.2 0.3^0.4

"d

Figure 3.4. Critical Mach number versus d/D,

485

4.00

2.00

• •

2.00

I

4.00

X

_J I

6. 00 8. 00

Figure 3.5. Plots of the supersonic pocket for M^ = 0.42,

486

4.00

2.00

•i^^

\ \ \ I

2.00 4.00 6.00 8.00

X

Figure 3.6. Plots of the supersonic pocket for M = 0,A5,

487

4.00r

2.00

«o^

2.00

I

4.00

X

6.00 8.00

Figure 3.7. Plots of the supersonic pocket M = 0.50.

488

(a)

4.0r

10

Speed 2.0

^ ^00^

LO

_ O

O
O
OO L

O
O
O
J Q

(b)

Speed

3,0
2.0

%Kg)

4.0 8.0 12.0

0X12/7T

Figure 3.8. Velocity as a function of angle: (a) on cylinder, (b)

slightly off cylinder — Mq ~ '^l*

p

489

Speed

B.OOr
2.50
2.00
1.50

1.00

.50f- O
O

O

O O

O

O

O

O

O

1

1

1

1

o

2.00 4.00 6.00 8.00 10.00 12.00

Figure 3.9. Velocity as a function of angle on the cylinder — full least
squares scheme with density modification — M = .5.

490

Speed

2.00

1.50

1.00

.50,

O
O

oo®cPo

UcP

O^

o
o

o

o

o
o

%

R)

o

2.00 4.00 6.00 8.00 10.00 12.00

Figure 3.10. Velocity as a function of angle slightly off cylinder — full
least squares scheme with density modification — M = .5.

491

2.00
1.50

Speed 1. 00

.50

O O ^
O O

O

O

O

O

o

o

o

o

1

1

1

1

2.00 4.00 6.00 8.00 10.00 12.00

Figure 3.11. Velocity as a function of angle half radius above cylinder —
full least squares scheme with density modification — M = .5.

492

4.00

3.00

Y 2.00

1.00

1

1

1

1

1

1.00 2.00 3.00 4.00 5.00 6.00 7.00 8.00

X

Figure 3.12. Supersonic bubble — full least squares scheme with density
modification — M = .5.

493

THE WEAK ELEMENT METHOD APPLIED TO HELMHOLTZ TYPE EQUATIONS

Charles I. Goldstein

Department of Applied Mathematics

Brookhaven National Laboratory

Upton, NY 11973

ABSTRACT

Helmholtz type boundary value problems are important in a variety
of scattering and diffraction problems. Standard numerical schemes
based on finite difference, finite element, or integral equation
methods are generally not well suited for these problems in the
"intermediate frequency range" since the oscillatory solution is not
accurately approximated by piecewise polynomials. In this paper, a
version of the weak element method is employed to numerically solve
these problems in two dimensions. This method consists of parti-
tioning the domain into small "elements" and locally approximating the
solution in each element by a sum of exponentials. These piecewise
approximations are joined together at interelement boundaries by
continuity conditions for certain functionals of the approximate
solution. The method is analyzed using a complementary variational
formulation. It is shown that the weak element method is considerably
more accurate than standard discretization methods when the solution
is adequately approximated locally by the exponential basis
functions. These results are validated by numerical experiments.

The submitted manuscript has been authored under contract DE-
AC02-76CH00016 with the U. S. Department of Energy. Accordingly, the
U. S. Government retains a nonexclusive, royalty-free license to
publish or reproduce the published form of this contribution, or allow
others to do so, for U. S. Government purposes.

495

1 . INTRODUCTION

It is the purpose of this paper to analyze and numerically
investigate the weak element method applied to Helmholtz type boundary

value problems in multi-dimensional domains. Scalar and vector

2
Helmholtz type equations, (A + K n)u = 0, with an appropriate

radiation condition and spatially dependent index of refraction, n,

are of importance in a variety of stationary wave propagation problems

occurring in acoustics, optics, seismology, and electromagnetic

theory. Since the solution is rarely known in closed form, it is

important to approximately solve these problems numerically in the

intermediate frequency range, where asymptotic methods can be

unreliable.

When applying typical discretization methods such as finite

difference and finite element methods as well as integral equation

methods, one is faced with the "resolution problem". This means that

in order to approximate the solution accurately when the wave number,

K, is not small, one must decrease the grid size, h, and hence solve a

prohibitively large number of linear equations. This problem arises

from the use, in the usual discretization methods, of piecewise

polynomial functions to approximate a highly oscillatory solution.

Methods for overcoming this difficulty have been developed in [1] and

[2] by combining the finite element method with functions satisfying

the desired oscillatory behavior. The method in [1] was developed for

one-dimensional problems. The method in [2] was designed to treat

multi-dimensional problems for which most of the propagation occurs in

a narrow angle band about a fixed direction.

496

An alternative approach for discretizing boundary value problems
is given by the weak element method developed in [3]. This method is
based on partitioning the domain into small subdomains (elements) and
approximating the solution in each element by a solution of a
localized approximation of the differential equation. These plecewise
approximations are joined together at interelement boundaries by
continuity conditions for certain functionals of the approximate
solution. See [4] and [5] as well as references cited there for a
discussion of related methods. In this paper we consider a version of
the weak element method in which the approximate solution consists of
piecewise exponential basis functions joined together at interelement
boundaries by imposing continuity conditions on the average values of
the approximate solution and its normal derivative. This method is
described briefly in Section 2 and in detail in [3].

In Section 3 we analyze this weak element method for a model
problem in a rectangle. The analysis employs a complementary
variational principle developed in [4] in connection with the Laplace
equation. Here we extend the arguments in [4] to a non-self ad joint
Helmholtz boundary value problem. We prove that when K h is
sufficiently small, the resulting discrete problem is well-posed and
the mean-square discretization error is of order 0(K h ) as h-»-0. This
is analogous to the situation for standard second order finite element
or finite difference schemes. We also show that when the phase of the
solution is adequately approximated locally by the exponential basis
functions, the weak element method is much more accurate than standard
discretization schemes as K increases. This is the main advantage of

497

the weak element method. Some techniques for approximating the phase
of the exact solution are described in [1] and [2]. In Section 4 we
demonstrate the results of some numerical experiments with the weak
element method. We summarize our conclusions in Section 5.

498

2. THE WEAK ELEMENT METHOD

In this section we outline briefly the weak element method
described in [3]. We employ the following notational convention.
Suppose that a=(a ,a ,...,a ) and b denote vectors with n components,
and $=(j5 ) denotes an nxn matrix whose ith column is {5 and whose jth
row is ^ . We denote the inner product of a and b by a"b and the norm
of a by |a|=(a*a) \ No notational distinction is made between row
and column vectors. Hence a in a$ is a row vector, but a in $a is a
column vector.

We consider the following differential operator acting in a

bounded domain D in the x=(x ,x-) plane with a piecewise smooth

boundary, 9D. Suppose that P and A are 2x2 matrices (P being positive

definite symmetric) and b and q are scalars. Let ii denote the outward

directed unit normal to D and let V=(9/ ,9/ ) denote the

9x^ 9x2

gradient. The linear elliptic operator L is defined in D by

L=-VPV+q, (2.1)

and the boundary operator B is defined on 9d by

B=n*AV+b. (2.2)

Before proceeding further we require the following additional
notation. Let H (D) denote a partition of D into N elements
(subdomains), {tt^}. We use a.(7r), j=l ,2, . . . , £(7r) , to denote one of
the £(Tr) smooth sides of the element ir. The vector

o(ir)=(a^(iT),a2(Tr),.. . .Oj^, (ti)) denotes the sides of tt oriented in a
counterclockwise manner about tt. A side a.(7r), which is incident to
another subdomain, it", is an interior side and is denoted by 0(11,1;').
Otherwise a.(ir) lies on 9D and is denoted by a'{T^). (See Figure 1 for
the case of rectangular elements.)

499

The area of tt (length of 0.) is denoted by liflClo.l). Let

9(tt) denote the smallest angle between the centroid, x , and any two

distinct vertices of ir. In order for the resulting system of linear

equations to be well conditioned, we assume that 9(Tr)>9 >0 for each

o

element ireII,,(D), where 6 is independent of N.
N o

We define localizations, L(it) and L(ir), of the operator L given

by (2.1) with respect to the element it as follows:

L(tt)=-V(P V)-(VP )-V+q (2.3)

000

and

L(Tr)=-V(P V)+q (2.4)

00

where P denotes P evaluated at x , etc. Finally, if u(x) is a smooth

(possibly vector-valued) function and o.^t:) is an arbitrary side

of ir, we define

and

-J ' J ' j(Tr)
u(a(Tr)) = (u(a^(T7)),u(a2(iT)),... ,u(a^, .(ir))),

02^^)

o^i.n

o^Ctt)

a^(^)

Figure 1

a^(^')

500

We are now ready to describe the weak element method employed
here to solve the boundary value problem Lu=0 in D, Bu=g on 3D. For
each element ir, let (S (x,ir), <S„(x,ii) , . . . ,/5 , .(x,7r) denote a linearly
independent set of solutions of the localized equation

L(tt)«5^=0 (2.5)

and define

KX, TT ) = (()) ^(x,Tr),(t)2(x, IT ),..., (j)^,,(x,TT)).

Our approximate solution on i: is now defined by the equation

w(x,7r)=^(x,7r)*a(rr), (2.6)

where the coefficient vector a(Tr)=(a. (it) ,a (it) , . . . ,a- , v(tv)) is
unknown.

Now suppose that a.(7r) is incident to ir' at the side
a.^(iT') and let a'(Tr) be a side of it on the boundary 9D. We impose the
following continuity and boundary conditions on w(x,t7):

w(a (TT))=w(a..(TT')), (2.7a)

(n^-pVw)(a^(Tr))=(n •PVw)(a..(Tr')) (2.7b)

on interior sides, where n. is the outward directed unit normal to

a . ( IT ) , and

(Bw)(a'(TT))=g(a'(TT)) (2.7c)

on boundary sides. Substituting (2.6) into (2.7), we obtain a system

of linear equations for the N vectors

a(TT^)=(a^(TT^),a2(TT^),.,. ,a^. . (tt^)) ,i=l , . . . ,N.

501

It is shown in [3] that the weak element approximation given by (2.6)
may be obtained by solving an equivalent smaller system of equations
for the average values of w on all sides, a.(-n), of the partition.
Remark 2.1 ; As described in [3], the weak element method can be
generalized as follows. In (2.7), we impose boundary (continuity)
conditions on the average value of the function (and an appropriate
derivative) for each boundary (interior) side, o.(tt). To generalize
the method we can replace the average value on a.(TT) by a set of
linear functionals on o.(Tr), denoted by <A (a. (it)) ,u>,m=l , . . .M. We
would then require M£.(Tr) local basis functions in each element. (For
example, these linear functionals might consist of the average value
of higher order moments of u on each side.) This could lead to higher
order methods than the method discussed in this paper for which M=l
and <A (o. (t^)) ,u>=u(o. (it)) .

For the sake of simplicity, in the remainder of the paper we
consider the special case in which each element tteII (D) is a rectangle
with sides parallel to the x ,x coordinate axes. As will be seen in
the next two sections, a key to the success of the weak element method
lies in the choice of the local basis functions, (j). (x,Tr) ,i=l , . . . ,4.
We employ an exponential basis defined as follows. Let x =(x ,x )
denote the center of it and define the unit vectors
e =(1,0) and e„=(0,l). We now set

/On / 0\ / o.

p^(x^-x^) _ p^{-x.^-x.^) _ -p^(x^-xp

(f)^(x,iT)=e ,(J)2(x,TT)=e ,<i>^(K,v)=e

~'^2^^2~^2^
and (f>, (x,ir)=e

) (2.8)

502

where p and p are chosen so that each (fi.(x,TT) satisfies (2.5). A
simple calculation yields

^f\'^Ceypl^)~ '\2=l,l' (2.9)

Basis functions analogous to those given by (2.8) and (2.9) can be

obtained by solving the equation L(i7)(|).=0 instead of (2.5).

The basis functions in (2.8) can be generalized as follows.

Define the unit vectors e^^=(cosa,sina) and e =(sina,cosa)

with 0<a<— . Now set
4

<J)^^(x,Tr)=e ,<|.2^(x,Tr)=e ^

> (2.10)

(Z-^°\ - -^ — /- o.

*3^(x,u)=e 1« 1" , and *,^(x..)=e ^^ 2a ^^ J
The constants p^^ and p^^ can be determined as before by substituting
(2.10) into (2.5). Note that a can have different values in different
elements. This can be useful when some knowledge is available
concerning the phase of the exact solution (see Remark 3.1 below).

The finite difference equations obtained using basis (2.8) were
derived in [3]. See [6] for a detailed investigation of the result-
ing finite difference formulas using both (2.8) and (2.10) and for
various aspects of the implementation of the method. The resulting
system of equations may then be solved for the unknowns,
a. (if^), j = l, . . . ,4,i=l, . . . ,N. The weak element approximation, w(x),
is obtained from (2.6). Hence we obtain w(x) at each point x in D
instead of only at nodal points. Observe that the resulting matrix is
highly sparse. Furthermore, the corresponding large system of

503

equations is nonself ad joint with indefinite symmetric part for
problems of the kind considered in this paper. The preconditioned
iterative method developed in [7] is well suited for solving this
system of equations. This iterative solver has not been implemented
in connection with the weak element method at the present time.

504

3. ERROR ANALYSIS

In this section we consider, for the sake of simplicity, the
following model problem:

(a) (-A-(K^+i6K))u=0 in D, ]

^ (3.1)

(b) u=g on 9D,

/

where D is the unit square, 6>0, and we assume that the solution

2
ueC (D). The term i6K is chosen to simulate a radiation condition as

in [8]. Furthermore, it is easily seen that this term ensures the
well-posedness of (3.1). We set K'=/K^+i6K and note that q=iK''
in (2.1) and b=l in (2.2). Furthermore, P(A) is the 2x2 identity
(null) matrix in (2.1) ((2.2)).

We shall employ the weak element method described in Section 2
with local basis functions given by (2.8) and (2.9). Hence we have a
partition of D,n^=nj^(D), into small rectangular elements,

TT^, i=l,...,N, such that the local basis functions defined on ir are

i

given by

+iK'(x^-xJ) +iK'(x2-xJ)

^ ' e . (3.2)

where (x^ ,^^ ) denotes the centroid of tt.. Denote the lengths of the
horizontal and vertical sides of tt. by h^"" and h^^, respectively, and
define

h=max max(h ,h„). Cq on

1 N

We shall analyze the discretization error using a complementary
variational formulation developed in [4] for the Laplace equation.

505

Before describing the variational formulation, we introduce some

additional notation. For a fixed element

±^\y let a ^=a (ir^) , j = l , . . . ,4, denote the four sides of . (see

Figure 1 above) and set 0(0. )=1 (-1) if a. . is to the right or top

of (to the left or bottom of) it.. If a =a., .^ is a common side

^ J > i J >i

of iT^ and 11^, veH (ir^) flH (tt^, and v.(v:) is the restriction of v
to 'iT.(Tr^), we define

(a) 6v Sp(a .)v +p(a., .^)vT for each interior
J » ■'-

side a. .=a., ., and

>

T =P(c^. .)v -p(a. . )g for each boundary

J > ■■■

(3.4)

J.i

(b) 6v

side a.

We next define some Sobolev and piecewise Sobolev spaces that are

important in the variational formulation. Suppose B=D. By H™(B), we

denote the space of functions v such that

||v||2 = lE,^ IId^vM^, <»,

^B) l"!^'"" "l2(B)

where m is a non-negative integer and D denotes a derivative of
order |a|. Let Hq(D) denote the closure of Cq(D) with respect to the
norm, | | | | .We define

h\d)

H™ E{veL'^(D):||v||^ ^= E „ Mv.ir <<»}.

- H (ir^)

H

m i N

We also define

2 N 9

for each veH

m

506

where the seminorm, { { > is defined by

|v|^ =E ||d"v||^ for each v£H"'(7r.).
h"(ti^) |ct|=m l'^Ctt^) ^

Finally, set

1^ 1 2
h; ={veH :(-A-(K +l6K))v =0 for each ir.en }.
K. 1 1 N

We now define the subspace H uH by

K.

E_ 1
H = ^^^ -^ has continuous normal

derivatives on 9ii . for each ir.en,.
1 1 N

Furthermore, we define the following bilinear form:

Aj^(v,w) = J^^/^ (Vv.Vw*-(K +i6K)vw*)dx Vv,weH , (3.5)

i

*
where w denotes the complex conjugate of w. It is easily seen that

the solution, u, of (3.1) satisfies the following variational problem:

Find ueH such that

(VP)

A^(u,v)=r (v) = I 6 g-5 — ds for each veH ,
K u ^ ^ on

o . . a
J,i 3,1

where the summation is taken over all element sides, a' ., contained

g

in 9D, ds denotes arc length, and -r— denotes the outward directed

on

normal derivative to 9D.

507

We discretize (VP) by defining a finite dimensional subspace,
S a y as follows. Let A denote the functions, ^ , defined on the

element sides a. , such that ^ is some constant, c. . on a. .. For

J,i h h ^'^ J'^

, ,h . .h - ^ h . „1 ^ . r 9v / . ,h ,

each V in A , let v in H^ satisfy -5 — = p(a. .)Tp on each

K dn J,l

9 ^

side a. . , where -^ — denotes the outward directed normal derivative

h
to a. . = cr.(Tr.) from it.. Hence v is the solution of a well-posed
J.i J 1 1

Neumann problem in each element. Let S consist of all such

functions v . By the construction of S , we see that S CH . We now

K- K

formulate our discrete variational problem.

Find u eS such that
K

.h/ h h\ _ / hx „ h „h
Aj,(u ,v )=r^(v ) Vv eSj,.

► (DVP)

Note that the weak element and finite element methods are based on

complementary variational principles in the sense that essential

boundary or interface conditions for one are natural conditions for

the other.

We next show that (DVP) is equivalent to the weak element method

described in the previous section. Suppose that u satisfies (2.7)

with local basis functions given by (3.2). In view of the definitions

of P and B corresponding to problem (3.1), it follows from (2.7) and

(3.4) that

.9u^

and

(a) S 6(-r — ) ds=0 for each interior side a. .

a . . dn . 1,1

> (3.6)

(b) ffl 6u ds=0 for each side a. ..
^(^. . o. . j,i

508

The local basis coefficients, a(7r. ) = (a. (tt. ) , . . . ,a, (tt. )) , are

determined in each it. so that (3.6) holds. It follows from (3.2) that

3^h "■

-g — is constant on each a.. Hence it follows from (3.2) and (3.6)(a)

J v» v»
that u eS . Furthermore, it is seen from (3.5), (3.6)(b), and

integration by parts that (DVP) holds. Conversely, it is easily seen

that if u satisfies (DVP), then (3.6) and consequently (2.7) (with w

replaced by u ) are also satisfied. The basis coefficients a(Tr.) may

be readily obtained as in [3].

It thus suffices to prove that (DVP) is well-posed and to

estimate the error, e =u-u , where u and u satisfy (VP) and (DVP),

respectively. To this end we first state the following result.

Lemma 3.1 ; Suppose that ^ satisfies the following boundary value

problem:

(-A-(K^-i6K))i|J=z in D, t(;=0 on 3D. (3.7)

Then

I'll 2 =CK||z|| 2 ' (3-8)

H^(D) L^(D)

where C is independent of K and z.

Note that we shall often use the same letter C to denote
different constants when there is no danger of confusion. Lemma 3.1
was established in [8] and shows how the norm of the resolvent
operator for (3.7) depends on K. This Lemma was also established with
the Dirichlet condition replaced by a radiation boundary condition on
part of the boundary. In such cases K is replaced by
K with 0<a<l, where a depends on various factors. See [8] for a

more complete discussion of these issues.

509

We are now ready to prove our error estimates. We first prove
the following Lemma using a duality argument as in [A].
Lemma 3.2 ; Suppose that u satisfies (VP) , u satisfies (DVP),
and e =u-u . Then there exists a constant, C, independent of K and h
such that

lle^ll 2 <CKh|eh|

Proof: First observe that

, |/_,e z*dx|

I I n I 1 ' ■' D '

L^(D) ZEC (D) " "l^Cd)

1 ^^ '"
Let \|;eHQ(D)flC (D) denote the solution of (3.7). For each vertical

(horizontal) element side, a. ., let ^-denote -r — (-r — ). It follows

J.i on 3x 3x„'

from (3.4), (3.5), (3.7), and integration by parts that

/pe^z*dx=/jje^(-A-K'^)/dx

(3.10)
h 9lL„^.h, h

h E 1^
Since e eH CH^ , it follows readily using (3.5) and integration by

parts that

'^(e\*).f.,/ (-i-K-^)eVdJ.J_j <(^ 6(|S-,_^ /.s-O.

h

-)

Combining this with (3.10), we deduce

h * ,- ^ r . h „ .*

/ e z dx=-J i 6e 8i|; ds. (3.11)

j,i j,i an

510

To estimate (3.11), we first divide each rectangular element,

IT., into two right triangles as shown in Figure 1

(with IT and tt' replaced by it. and ir'). For simplicity, consider the

triangle containing sides a and a, .. Denote this triangle by t

1,14,1 i

and set ^.='|'| • Set

i

'1,1- ^»i,,V^ -^ "4,1- Jx^x^'^"'- "•'"

where \a. , | ,j=l or 4, denotes the length of side a. . Define
J'^ J,i

^I^v^,i-l•^^,i-2

and note that

j,i

Since ueC(D), it follows from (3.6)(b) that

{>^ 6e ds=0 for each side a. . (3.14)

Using (3.14) and a scaling argument as in [4], it may be seen that

^^ |6e^ |2ds<Ch(|e^| ^ ^\,^\2 ^^

j,i j,i hVtt^) H^Tip

where C is independent of h and tt (and j = l in Figure 1).

(3.15)

9'1'i ^^■

Note that — — — is constant on a. . Combining this with

(3.14), we obtain

511

^r, ^^r, "5 ds=(^ 6e -r ds.

o, . a. . dn o. . a. 9n

(3.16)

It follows from Schwarz ' inequality and (3.15) that

^K. .< .l!^=isil«''o. II

J.i J.i

. , ,2, J ' 3n I
J,i L (a )
J > 1

^<"j.i>

>(3.17)

J/9 , 1 h

H.'

.<Ch'2(|e''| .|e^ Jll-ji-ll

H (TT^) H (<)

^<°J.l'

Using an argument in [4] (see Lemma 2.2.6) and (3.13), we deduce

X

3n " 2 <^^^\\\ 2 ^Ch/2||^|| ,
L^(a ) ^ H^(t.) H^TT.)

J » -*- 1 1

(3.18)

Combining (3.8) with (3. 16)-(3. 18) and the Schwarz inequality, we
conclude that

lf,j K. .< .|^is|<CKh||z|| 2 le^l ,. (3.19)

J.i j,i

L (D) ^1

Finally, we combine (3.9), (3.11) and (3.19) to complete the proof.
Q.E.D.

We now prove our main result.

2
Theorem 3.1 : Suppose that u satisfies (3.1) and ueC (D). Then for

K h sufficiently small,
there exists a unique solution u satisfying (DVP) and

(3.20)

512

I h,2 ^ < C inf , ., h,2 , ^„2 , , h||2 . (3.21)
H^ ^ ^h H^ L^(D)

Furthermore,

/" L^(D)- W^CD)

The constant, C, in (3.21) and (3.22) is independent of K, h, and u.
Proof: First, assume that u exists. Using (3.5), we immediately
obtain

le^l^ <|A^(e^,e^)| + |(K^i6K)lMe^||% . (3.23)

1^ - ^ L^(D)

n

Employing Lemma 3.2 and condition (3.20), we may "kickback" the last
term in (3.23) to obtain

|e I ^ <C|\(e ,e )|.

In view of (VP) and (DVP), the last estimate yields

|e^|^ ^ <C|A^(e'',u-v^)| for each v^ in sjj. (3.24)

It follows readily using (3.5) that

|A;;(e\u-v^|<|e^| Ju-v^| ^+(K^i6K) | |e^| | ^ Mu-v^|| ^

^l^ L^(D) "^ jjl^ L^(D)

513

for arbitrarily small ti>0. Combining this estimate with (3.24) and
again applying Lemma 3.2, (3.20), and a "kickback" argument, we obtain
(3.21).

We next prove (3.22). In view of (3.24), we construct a
function v in S satisfying

|A;;(e\u-v^|<Ch2||u||2 -Hnle^l^ (3.25)

for ri>0 arbitrarily small. Suppose v in II has sides

^^ j.j = l.«".4. We define four constants c .,..., c. such that

K. /j,i'^^= K. P(o )^s,i^l,...,A, (3.26)

J.i J,i '' i

where n is the outward directed unit normal to 9t7 . It is easily

seen using the Mean Value Theorem for integrals that for each

j=l , . . . ,4, we have

c. .=p(a. .)|^(P. J
j,i J,i'3n^ J,i

for some point P. .ea. .. It follows readily using the Mean Value
Theorem for derivatives that

ll%-''<".y=.,ill,2<_,i-(^„^Jli%ll^^_->"^ (3...,

j = l,...,4. We now define i); in A by il) I =c. , for each side a

514

It follows immediately from (3.26) and (3.27) that for each a. ,, we have

(a) i -;; — ds=^ p(o. . )ip ds and

^a . , dn. ^c . . J ,1

j.i i

3,1

(b) I||^-P(a .)/|| 2 <Ch'l|u|| 2 ■

(3.28)

To construct v satisfying (3.25), we solve the Neumann problem for
equation (3.1)(a) in each u with Neumann data given by
p(a. .)il; on each side a. ., i = l,...,4. We denote the solution by
V . Set v =v. in each it. and note that v eS . We now employ integration
by parts to deduce

A^(e^,u-v^)=S ^g^ e^— (u-v^)*ds. (3.29)

i i ^

It follows immediately from the construction of v and (3. 28) (a) that

,^ ^ — (u-v ) ds=0 for each ir.
dTT.dn. 1

1 1

In view of this, we may replace e on the right hand side of (3.29) by

e^=e^-c^ in tt^ (3. 30) (a)

with c =-rr — r^„ e ds, so that

ffl. e.ds=0 for each it.,
'^dn, i 1

(3.30)(b)

Applying (3.28)(b), (3.29) (with e^ replaced by e^ on the right side), and
the Schwarz inequality, we obtain

515

\^ie\u-.'^)\<J t^ llejil 2 |||^-p(, ),^||

^ L (0. .) 1 ^'^ L'^co. .;

(3.31)

W^(D) "- ^ "- L^Ott.)

1

To estimate | |e. | | , we map tt . onto the unit square, ti,

and employ the following well-known estimate:

||w||^ <C(|w|^ +|^„ wds|^) for each w in H^(Tr).
H^tt) h\tt) '^^

As in [4], we combine this estimate with (3.30)(b) and map it back onto

IT. to obtain

1

|e;;|| <Ch/2|eh| =Ch/2|e^| ,
L^(3Tr^) ^ H^TT^) H^TT.)

(3.32)

using (3.30)(a) in the last step. We now combine (3.31) and (3.32) to
conclude that for arbitrarily small ri>0:

.h, h h^i ,„, 2

N

|A;:(e".u-v")|<Ch^ Z^^\\u\\ 2 |e"|

W„ (D) H"(TT^)

(3.33)

< J^ (Ch^||u||2 +n|e^|\ ).

_2
Estimate (3.25) now follows from (3.33) since N=0(h ),

and (3.25) with n>0 sufficiently small, we deduce

Combining (3.24)

|e"| h <Ch||u|| ,

(3.34)

516

Estimate (3.22) now follows from (3.34) and Lemma 3.2. Finally, to prove

that (DVP) is well-posed, it suffices to prove uniqueness since

S is finite dimensional. If g=0 in D, then u=0 in D since (3.1) is well-

posed. Hence it follows from (3.22) that u =0 in D and we have proved that

(DVP) has a unique solution. Q.E.D.

Remark 3.1 : Typically, the solution of (3.1) satisfies ||u|| =0(K ).

W (D)

■L TO

Hence it follows from (3.22) that | |e || =0(K h ). This is analogous

L^(D)
to results obtained for standard second order finite element or finite

difference schemes (see [8]). However, it follows from (3.21) that the
weak element method is clearly superior for moderate to large values of K
when the oscillatory behavior of the solution is well approximated in each
element by the local basis functions, assuming the "stability" constraint
(3.20) holds. This constraint also occurs in connection with standard
discretization schemes. Our numerical results indicate that this stability
constraint does not cause serious computational problems for the weak
element method when the oscillatory behavior of the solution is well-
approximated by functions in S^,. We shall see in Section 4 that in such
cases the discretization error is quite small even when K h is large.
Remark 3.2 ; It follows from the previous remark that the main
computational advantage of the weak element method occurs when the
oscillatory behavior of the solution is well-approximated by functions
in Sj^. The determination of this oscillatory behavior can be difficult for
realistic physical models. This question was investigated in [1] and
[2]. In [1], asymptotic methods were employed in connection with a one-
dimensional scattering problem. In [2], multi-dimensional models were

517

treated for which it is known that most of the propagation occurs in a
narrow angle band about a fixed direction. This condition is closely
related to the "paraxial approximation" and holds in a variety of
application areas.

Remark 3.3 ; The weak element method described in Section 2 can be extended
in various ways (see Remark 2.1). Alternatively, the variational
formulation described in this section can be generalized by employing
higher order approximating subspaces S^,. See [4] for a detailed discussion
of this in connection with the Laplace equation. Furthermore, more general
boundary value problems can be treated than (3.1). This includes more
general domains, variable coefficients, and radiation boundary
conditions. We intend to investigate some of these questions in the
future.

518

4. NUMERICAL ANALYSIS

In this section, we demonstrate the effectiveness of the weak
element method described in Section 1 for simple two-dimensional test
problems whose solutions are known in closed form. Our measure of
error is given by

where u(u ) is the exact (approximate) solution, | | | | denotes

aw

the discrete mean-square norm, and D is a rectangle in either
Cartesian or polar coordinates. D is partitioned into rectangular
elements as described in Section 2 such that the grid points are
equally spaced in each direction. We denote the number of intervals
in the x and x„ directions by N and N , respectively. The
differential operator is given by (3.1)(a) with 6=0.

Our boundary condition for the first two examples is the
Dirichlet condition, (3.1)(b), although we have obtained analogous
results for various combinations of Dirichlet, Neumann, and impedance
boundary conditions. In Example 3, we consider the Helmholtz equation
in polar coordinates in the exterior of the unit circle with a
radiation boundary condition imposed on an artificial outer
boundary. Our main purpose in all of these examples is to evaluate
the discretization error for different values of N , N„, and K. The
calculations were performed on a CDC 7600 at Brookhaven National
Laboratory. The system of equations were solved using a standard
conjugate gradient iterative method applied to the normal equations as
well as a direct solver based on Gaussian elimination. Both methods

519

resulted in essentially the same discretization errors. It is

expected that more recently developed preconditioned iterative

methods, such as that discussed in [7], would be considerably more

efficient.

Example 1 ; For our first series of numerical experiments, we assume

that D is the unit square and choose Dirichlet boundary conditions

such that the solution is given by

u(x ,x„) = sin K(x.cos a + x„ sin oc), < a < — . (4.1)
We employ the weak element method with local basis functions given by
(3.2) (with K'=K). Our results are demonstrated in Tables lA - D with
Nj^ = N2 = N = h . We have also employed the five-point finite
difference scheme in this and the following example although it is not
necessary to demonstrate the results obtained using this scheme. It

suffices to observe that, as expected, this finite difference scheme

2
has convergence rate 0(h) as h+O with K fixed. (Our numerical

results indicate that this is also the case for the weak element

method.) Furthermore, the five-point scheme is not accurate when

Kb = I > 1.

It is readily seen from (3.2) and (4.1) that when a=0 or a=^, the

solution in each element may be expressed as a linear combination of

local basis functions. Hence we would expect the weak element

approximation to yield the exact solution, except for accumulated

roundoff errors. This is validated in Table lA for N=4 and various

values of K. On the other hand, it follows from (3.2), (3.21), and

(4.1) that the more a differs from and -^ , the less effective the

weak element method should be with this basis (see Remark 3.1). In

Tables IB - D, we consider various values of N and K with

520

IT IT , IT

a=-rTp:. -TT, and -r-, respectively. We see from Table IB that when

150' 25'

IT

iTo'

ex = -T-^, the weak element method is accurate even when Kh = 16. From

77

Table IC we see that for a = — , the method yields accurate results

when Kh = 2 and hence is more effective than the five-point finite

difference scheme. On the other hand, when a = — we see from Table ID

o

that the method does not yield accurate results when Kh > 1. We have
observed that in cases such as this for which the phase of the
solution is not sufficiently well approximated, there is no advantage
in using the weak, element method instead of a standard discretization
scheme.

521

Table lA
(N=4)

Table IB

a=

=0

a

=

IT

2

K

E2

E2

1

3.9

X

10-13

3.8

X

10-13

2

1.7

X

10-13

1.9

X

10-13

4

3.8

X

10-13

7.0

X

10-13

8

2.9

X

10-1^

1.7

X

10-12

16

1.2

X

10-13

6.1

X

10-12

32

3.4

X

10-1^

1.3

X

10-11

64

5.2

X

10-13

2.5

X

10-11

128

6.8

X

10-1^

4.9

X

10-11

256

2.1

X

10-1^

9.9

X

10-11

512

1.9

X

10-12

2.0

X

10-10

1024

1.5

X

10-13

4.0

X

10-10

2048

1.2

X

10-13

8.1

X

10-10

K

N

E.

I

1

4

5.8

X

10-5

1

8

1.6

X

10-5

1

16

5.2

X

10-^

2

4

2.3

X

10-^

2

8

6.2

X

10-5

2

16

2.0

X

10-5

4

4

8.7

X

10-^

4

8

2.5

X

10-^

4

16

8.2

X

10-5

8

4

3.5

X

10-3

8

8

9.9

X

10-^

8

16

3.3

X

10-^

16

4

1.2

X

10-2

16

8

4.3

X

10-3

16

16

3.6

X

10-3

32

4

2.6

X

10-2

32

8

1.3

X

10-2

32

16

5.4

X

10-3

64

4

4.9

X

10-2

64

8

2.7

X

10-2

64

16

1.5

X

10-2

128

4

9.7

X

10-2

128

8

6.9

X

10-2

128

16

5.1

X

10-2

256

4

1.9

X

10-1

256

8

1.0

X

10-1

256

16

5.6

X

10-2

522

Table IC

Table ID
(a=g)

K

1

1

1

2

2

2

4

4

4

8

8

8

16

16

16

32

32

32

64

64

64

4
8

16
4
8

16
4
8

16
4
8

16
4
8

16
4
8

16
4
8

16

3.7 X
1.0 X

3.2 X

1.4 X
3.9 X

1.3 X

6.0 X

1.7 X
5.6 X

2.3 X

6.5 X

2.1 X

1.1 X

8.6 X

1.2 X

1.8 X
1.2 X

5.9 X
3.1 X
1.8 X

1.4 X

10

10

10"

10

10

10

10

10

10

10"

10

10

10

10

10"

10

10

10

10

10

10'

-4
-4

-3

-4
-4
-3
-3
-4

-3
-3
-1
-2

-1
-1
-2
-1
-1

K

N

E2

1

4

2.1

X 10"^

1

8

5.6

X lO-'^

1

16

1.8

X 10"^

2

4

8.2

X 10"^

2

8

2.2

X 10"^

2

16

7.5

X 10"^

4

4

5.3

X 10"^

4

8

1.3

X 10"^

4

16

3.9

X 10"^

8

4

1.0

8

8

6.9

X 10"2

8

16

1.9

X 10"2

16

4

1.3

16

8

1.2

16

16

1.5

Example 2 ; For our next class of problems, we consider solutions of

the form

/'~2 2
■ I- I- ^ K -L X ,L=1,2,... , (4.2)

with Dirlchlet boundary conditions on the unit square. Our local

basis for the weak element method is again given by (3.2). For Kh«l,

the convergence rate is again 0(h). However, for Kh»l, the weak

523

element method behaves differently for this problem than for the

previous example. The reason for this is that the x, dependence of u

is independent of K. Suppose that K»L in (4.2), so that

u(x. ,x )'^sinLx cosKx„. Hence the X2-dependence of u may be reproduced

almost exactly by the basis functions for K large and the accuracy

will be almost independent of the number of yi.^ grid points. We are

thus left with approximating sinLx by constants, yielding an

0(N ) order approximation to u that is independent of K for large

K. We illustrate typical results in Table 2 for L=l, K=128, and

various values of Nj^ and No. •

Table 2
(L=l, K=128)

Nl

N2

]

h

8

9.5

X

10-2

16

4.9

X

10-2

32

2.5

X

10-2

64

1.3

X

10-2

128

6.4

X

10-3

8

2

9.3

X

10-2

16

2

4.8

X

10-2

32

2

2.4

X

10-2

64

2

1.2

X

10-2

128

2

6.1

X

10-3

8

4

9.3

X

10-2

16

4

4.8

X

10-2

32

4

2.4

X

10-2

64

4

1.2

X

10-2

128

4

5.8

X

10-3

524

Example 3 ; The final problem we consider is one treated in [9] using
an integral equation approach combined with a finite difference method

for K not too large. Introducing polar coordinates, (r,e), the

2
problem consists of solving the Helmholtz equation, (A+K )u=0 in the

exterior of the unit circle, subject to the boundary conditions

2
u(x)=x on r=l and the radiation condition

Ou/8r)-iKu=o(r"^^^) (4.3)

for r large. The solution of this problem is given by

x^ H^^^^(Kr) sineH^^^^Kr)

where H denotes the Hankel function of first kind and order 1.

In order to determine the discretization error due to applying
the weak element method to this problem, we replace the right-hand
side of the radiation condition (4.3) by the exact value obtained by
applying (9/8r)-iK to (4.4) on a circle of radius R>1 and denote this
function by g(R,9). Employing polar coordinates and appropriate
symmetry conditions on u(x), we obtain the following boundary value
problem for u(x):

u=sin6 on r=l,
(3u/3r)-iKu=g(R,e) on r=R, / (4.5)

(3u/90)=O on e=ii/2 and
u=0 on 0=0,
where D denotes the domain l<r<R,0<9<TT/2.

Problem (4.5) may be placed in the general framework of (2.1) by
replacing the (x ,x„) coordinates by (r,e) coordinates, so that

525

V=(9/9r, 8/96). Hence the domain D is a rectangle, the matrix P is
given by

^ ''O l/r-"'

2
and q=-K r in (2.1). Since (VP) =(1,0)7^0, we simplify the problem by

making the transformation

u(r,e)=r"^/^v(r,e).

We now obtain the following boundary value problem for v:

(_Vp'V+q')v=0 in D, ^

v=sin6 on r=l,

1^ - (iK+|^)v=R^^^g on r=R,

■gg=0 on 8=2 and
v=0 on 8=0,

(4.6)

where

P'=(^ ° „) and q'=-(K^-H^).
Or 4r

Hence (VP') =0. Using (4.4), we see that the solution is given by

,,„ r^^^sin0H/^^(Kr)

v(r,e)=r'^^u(r,8)= ^-y^ ,

H '^'^K)

(4.7)

We now apply (2.8) and (2.9) to obtain the following local basis
functions:

±i(K^+l/4r ^)^^^(r-r ), ±ir (K^+l/4r ^)^''^(e-8 ). (4.8)
e o o e o o o

We also note the following asymptotic representation of H^ (Kr)
(see [10]):

526

H/'>(K.):(^)'/2e^<>^-^"/« %' '-"'''i^' 3 (4.9,
'- ^' j-0 J!(21Kr)Jr(-j4)

for Kr large and J>1, where T denotes the gamma function.

If we compare (4.7)-(4.9), we see that the r-dependence of
v(r,e) is accurately reproduced by the basis functions for Kr large.
This is analogous to the situation in Example 2. In Table 3A, we
examine the error, E2, for R=2 and different values of K and N=N,=N2,
where Nj^(N2) is the number of subintervals in the r(6) direction. For
— small, the weak element method again behaves analogously to the
five-point finite difference scheme. On the other 'hand, E2 is nearly
constant for — large and N fixed as K increases.

Furthermore, we have observed that for larger values of R the

K K

errors are about the same as for R=2 when — is large. When — is

small, accuracy is destroyed by the coarse grid in the r-direction.

This can be remedied by using a graded mesh in which the r-grid sizes

are systematically increased as r increases (see [11]). We illustrate

the high frequency behavior in Table 3B, where K=R=128. In this case

the grid sizes in the r-direction are quite large. We observe that E2

is essentially constant when N2 (the number of intervals in

the 9-direction) is fixed and Nj^ varies. Furthermore, the error with

respect to Q is of order 0(N ). The explanation of these numerical

results is the same as that given in example 2 (i.e., the B-dependence

of v(r,e) is approximated locally by constants).

527

Table 3A
(R=2)

Table 3B
(R=128, K=128)

K

N

E2

1

8

8.9 X

1

16

3.7 X .

2

8

1.2 X

2

16

3.9 X

4

8

2.1 X

4

16

5.9 X

8

8

4.0 X

8

16

1.1 X

16

8

2.6

16

16

2.1 X

32

8

1.7 X

32

16

4.6

64

8

8.0 X

64

16

9.2 X

128

8

9.3 X

128

16

4.2 X

256

8

8.9 X

256

16

4.7 X

512

8

1.0 X

512

16

4.6 X

1024

8

9.5 X

1024

16

4.7 X

2048

8

8.8 X

2048

16

5.2 X

10

10

10

10

10

10

10"

10

-3
-3
-2
-3
-2
-3

-2

10
10

-2
-1

10
10
10

10
10

-2
-2
-2
-2
-2

10

10

10"

10

10

10

10

-2
-1

-2
-2
-2
-2

h

N2

^2

1

8

9.1

X

10-2

2

8

9.1

X

10-2

4

8

9.3

X

10-2

8

8

9.1

X

10-2

1

16

4.7

X

10-2

2

16

5.0

X

10-2

4

16

4.7

X

10-2

8

16

4.7

X

10-2

1

32

3.1

X

10-2

2

32

2.4

X

10-2

4

32

2.4

X

10-2

8

32

2.4

X

10-2

1

64

1.3

X

10-2

2

64

1.3

X

10-2

4

64

1.3

X

10-2

8

64

1.3

X

10-2

528

5. CONCLUSIONS AND COMMENTS

We have analyzed and tested a version of the weak element method
developed in [3] in connection with the Helmholtz equation.
Mathematical models of this kind occur in various scattering and
diffraction problems. Standard discretization schemes based on finite
difference, finite element, or integral equation methods are not well
suited for these problems when the wave number, K, is not small, since
piecewise polynomials are not good approximations to the oscillatory
solution. On the other hand, the weak element method is based on
piecewise exponentials that satisfy a localized approximation to the
differential equation.

We have proved that the particular weak element method outlined
in Section 2 with mesh size h has a convergence rate of order 0(h ) as
h-^0 for fixed K. Our analytic results also indicate that the method
yields a good approximation to the solution, u, when the oscillatory
behavior of u is well approximated by the local basis functions. The
proof is based on a complementary variational principle developed in
[4] in connection with the Laplace equation. It is expected that this
proof can be extended to more general boundary value problems and
higher order weak element methods.

We have also validated our theoretical results with respect to
test problems for which the solution is known in closed form. We have
seen from these examples that the weak element method offers no
advantage in general compared to the five-point finite difference
scheme. However, our theoretical and numerical results demonstrate
that the weak element is considerably superior for moderate to large K

529

when the oscillatory behavior of the solution is adequately
approximated locally. For general variable coefficient problems, this
oscillatory behavior will vary in different parts of the domain.
Consequently, an important practical area of investigation is the
development of methods for determining locally the approximate
oscillatory behavior of the solution for large K. This was done in
[1] in connection with a one-dimensional scattering problem using
asymptotic methods. This was also done in [2] for multi-dimensional
propagation models for which most of the propagation occurs in a
narrow angle band about a fixed direction. The use of error
estimators and adaptive discretization methods might also be useful in
determining appropriate local basis functions.

ACKNOWLEDGMENTS

The author wishes to express his gratitude to Dr. M. E. Rose for
several stimulating discussions during the course of this work. The
author is also grateful to H. Berry for his help in the preparation of
the numerical results in Section 4.

530

REFERENCES

[I] A. K. Aziz, R. B. Kellogg, and A. B. Stephens, "A two point
boundary value problem with a rapidly oscillating solution," to
appear.

[2] C. I. Goldstein, "Finite element methods applied to nearly one-
way propagation," J. Comp. Phys. , to appear.

[3] M. E. Rose, "Weak element approximations to elliptic

differential equations," Numer. Math. , 24, 185-204, 1975.

[4] I. Babuska, "The method of weak elements," Tech. Note BN-809,
Inst. Fl. Dyn, and Appl. Math., U. Maryland, 1974.

[5] J. Greenstadt, "Cell discretization," in Conf. on Appl. of Num.
Anal. Lecture Notes #288, Springer, Berlin, 1971.

[6] C. I. Goldstein and H. Berry, "A numerical study of the weak
element method applied to the Helmholtz equation," BNL Report
No. 50746, 1977.

[7] A. Bayliss, C. I. Goldstein, and E. Turkel, "The numerical
solution of the Helmholtz equation for wave propagation
problems in underwater acoustics," Comp. and Math, with Appl.,
11, No. 718, 655-665, 1985.

[8] A. Bayliss, C. I. Goldstein, and E. Turkel, "On accuracy

conditions for the numerical computation of waves," J. Comp.
Phys. , 59, 396-404, 1985.

[9] D. Greenspan and P. Werner, "A numerical method for the

exterior Dirichlet problem for the reduced wave equation,"
Arch. Rat. Mech. Anal. , 23, 288-316, 1966.

[10] I. S. Gradshteyn and I. M. Ryzik, Table of Integrals, Series,
and Products, Academic Press, New York and London, 1965.

[II] C. I. Goldstein, "The finite element method with nonuniform
mesh sizes applied to the exterior Helmholtz problem," Numer.
Math. , 38, 61-82, 1981.

531

THE LOCAL REDISTRIBUTION OF POINTS ALONG CURVES
FOR NUMERICAL GRID GENERATION

Peter R. Eiseman

Department of Applied Physics and Nuclear Engineering

Columbia University

New York, NY 1002?

ABSTRACT

A methodology is established to cluster points along curves in a
manner which does not change the existing pointwise distribution out-
side of a specified region containing the cluster. In each instance,
points are pulled from the perimeters of the region towards the clus-
ter center. The result is a smooth expansion from each end followed by
a compression into the center. Altogether, this represents a local
redistribution of points which can be employed either interactively or
automatically.

This work was supported by the US Air Force Grant AFOSR-82-01 76B and the NASA
Langley Research Center Grants NAG1-4Y9 and NAG1-H27.

533

INTRODUCTION

When a pointwise distribution along a curve is acceptable every-
where except in certain local regions, the capability to redistribute
points only in those regions becomes important. Our objective, here,
is to create a framework from which methods for the desired local re-
distribution can be developed. This is done by forming elementary
operations which are then applied in succession. Each operation
smoothly forms a single cluster about a point by attracting only near-
by points: the pointwise locations beyond a specified distance on
either side remain unchanged. As such, the action occurs in an in-
terval where both the endpoints and the internal cluster point remain
fixed: the other points move in from each side while maintaining a
globally smooth variation in pointwise spacing.

Upon application, a new pointwise distribution is created from an
old one and differs from it only in the chosen interval. The old or
"previous" distribution is always viewed as a mapping from a uniformly
distributed independent variable to the curve. This variable is often
referred to as just the existing parameterization for the curve. With
a finite number of points, a uniformly spaced grid along the paramet-
ric interval is mapped onto a grid along the curve. The new distribu-
tion is simply constructed by composition whereby we first map a new
uniformly distributed parametric interval onto the old one and then
apply the old map to the result. In terms of grids, a uniformly
spaced grid on the new parametric interval is mapped onto the old par-
ametric interval to produce a distribution there which is non-uniform
in some local subinterval. On that subinterval, the application of

534

the old mapping is accomplished with the aid of local interpolation.
Off of that subinterval, the points coincide with the old parametric
locations and no interpolation is required.

At each stage, there is a progression from old to new correspond-
ing to the application of an elementary operation. As noted above,
each such operation can be generated from a local reparameterization
which is just a mapping between old and new parameters. The actual
construction can be done in either forward or backward directions by
the use of weight functions. The forward direction is from new to old
while the backward is from old to new.

WEIGHTS AND TRANSFORKATIONS

With the view of larger masses pulling more strongly to a center
of gravity, weights are most commonly thought of as being more strong-
ly attractive when they are large. For the application to distribu-
tion functions, this view means that points are more strongly at-
tracted to locations of large weight. In correspondence, the point-
wise spacing must then shrink to adjust to a large weight. The most
simple way to have this happen is to make the spacing vary in an in-
verse proportion to the weight. In terms of changing the spacing in
the old parametric interval, we must then force the product of the
weight and the desired spacings to be equal to a constant. When the
same interval is taken for the old and new parameters, that constant
is just the increment from a uniform spacing. In our development, we
will always let s denote the old parameter and t denote the new param-
eter. In this notation and with our assumption of the same interval.

535

the forced condition is given by

dt = w ds (l)

and is called an equidistribution of the weight w since equal amounts
of weight must appear in each interval ds. For grids, the total
weight between every pair of points is then the same.

To construct the elementary operations of local clustering, we
recall the basic requirements that the cluster center and the interval
endpoints must remain unchanged. For the weights, these requirements
become integral statements; namely, that the integrals of wds and ds
are the same over both of the intervals from the cluster center to the
endpoints of the cluster region. Noting that uniform spacing would
occur if w = 1 , deviations therefrom are responsible for non-uniform
spacing and can be represented as a function f which is added to the
unity of uniformity in the weight to get w = 1+f. In terms of f, the
preservation of cluster center and endpoints results when the integral
of f vanishes over each of the two intervals above. To define these
intervals in a clear way, a zero subscript will be employed for the
center while a minus and a plus will be used as subscripts to indicate
the endpoints in negative and positive directions, respectively, from
the center. In this notation, the preservation condition means that
new t and old s must satisfy t_ = s_, tg = Sq, and t^ = s^ or that the
function f which gives variations from uniformity must integrate to
zero both from s_ to Sq and from Sq to s^..

To obtain a maximal amount of control over shape, such a function
is best created from a piecewise polynomial construction. The sim-

536

plest of these constructions is accomplished with two adjoining line
segments for each of the two intervals. This is depicted in Fig. 1
where it is clearly evident that the first segment from either end

Figure 1 : The function which must be
added to unity to form a weight for Eq. 1

must lie below the axis to create negative areas which are enough to
balance the positive area caused the linear rise to the positive value
at the center. The center value determines the intensity of the clus-
tering: the sum with unity gives the weight Wq, and hence, the min-
imum spacing dt/WQ. In compensation for the smallest spacing at the
center, the spacing must first increase and then start to decrease.
This starts from each endpoint spacing and linearly increases to a
maximum at the end of each segment below the axis. Upon forming the
weight w with an addition to unity, the maximum spacing on each side
is given by dt/w with w evaluated at each corresponding end. Aside
from the obvious limitation on the maximum spacing imposed by the

537

total interval length, there is the basic limitation that the weight w
must be positive: negative weights flip the incremental intervals;
thereby, causing a singularity in the mapping and a folded grid. As a
consequence, there is then a limitation also on the intensity of clus-
tering at the center. This is caused by the required balancing of
positive and negative areas in Fig. 1.

Obeying the intensity limitation, the elementary clustering oper-
ation is obtained by a direct integration of Eq. 1 with our weight.
The consequent mapping is then expressed with the new parameter t
given as a monotone function of the old parameter s. In correspond-
ence with the linear segments of the integrand, t is expressed as a
piecewise quadratic function of s. To apply the mapping, a uniformly
distributed t must produce the desired non-uniformities in s which in
turn are sent to the curve by using the old curve mapping. This is
just the composition of going from t to s and then to the curve. By
construction, however, we go from s to t which Is backward. This
means that t(s) must be inverted to obtain s(t) which is forward rath-
er than backward and thus can be used directly. Fortunately, in this
piecewise quadratic case, the analytical inversion is possible and is
somewhat simple. Since it contains radicals, it is not as simple as
the original backward construction.

With the motivation towards more simplicity and higher levels of
clustering intensity, we shall consider forward rather than backward
constructions. To accomplish this, we must invert our thinking about
weights, and thereby, have points attracted to locations where the
weight is smaller rather than larger as would have been expected when

538

compared to the center of gravity shifts for masses. In terras of the
piecewise-linear construction depicted in Fig. 1, the inversion re-
sults in a rigid reflection about the horizontal axis and a relabeling
of that axis to be for the new parameter t in place of the old s.
This is displayed in Fig. 2 and as earlier is added to unity to form
the weight w = 1 +f which is now used in

ds = w dt

(2)

For notational consistency, the new parameters t_ = s_, tg = Sq. and

Figure 2 : The function which must be added to unity
to form a weight for the forward mapping with Eq. 2

t+ = s+ are used. The equalities also result from the rigidity of the
reflection.

539

Rather than derive the algebraic formulation directly from t_ to
tg and then from tg to t^, we first re-examine the basic constraint
which led to the equalities above; namely, that the integral of f
over each interval vanishes. This constraint must be satisfied not
only by the function displayed in Fig. 2 but also by any function
which is to be employed for the same purpose. To begin our re-exam-
ination, we first note that the two integrals still vanish, if we
rigidly translate the function along the t-axis. The translation is
just the transformation from t to t+c from some c. Moreover, we also
note that the vanishing is preserved under a constant dilation or
contraction of either vertical or horizontal axes. These are just
transformations which scale an axis by scalar multiplication. In the
horizontal case, it is the transformation from t to at for some a.
The effect of either transformation is to multiply the vanishing
integral by a finite constant, and thereby, preserve the vanishing.
In more formal terms, the constraint is invariant under the groups of
transformations for translation and scaling. As a practical conse-
quence, we can derive our algebraic formulation with standard condi-
tions for height and interval and then apply the transformations to
get the formulation for any other conditions that we wish. This also
means that the same derivation can be used for the intervals on each
side of Iq-, and consequently, reduce the complexity of derivation by
half. A further reduction comes from selecting the unit interval and
a unit height since the arithmetic will be simplified.

With the unit lengths for our standard conditions, we are led to
consider the function shown in Fig. 3 where the juncture point loca-

540

tion X = a is arbitrary. From a given downward unit f (1) = - i and

a

the requirement for equal areas above and below the x-axis, we find
that f^ must cross the x-axis at 1/(2-a) and have a value of 1-a at
a. This function is then uniquely determined by the value at a which
is also indicated in the figure. From the figure, the algebraic form-
ulation is directly seen to be

(a, 1-a)

(1,-1)

Figure 3 ; A standard function for the
construction of piecewise linear weights

541

(1-a)(-] for < X < a

f (x) = •< (1-a) - (2-a)(^] for a < X < 1

"\

otherwise

(3)

As a matter of interpretation, a represents the location of maxi-
mal spacing while the intersection point 1/(2-a) represents the loca-
tion where the spacing starts to decrease beyond the original uniform
spacing. This means that the desired impact of clustering becomes
significant only after the intersection point since we must first re-
cover from the large spacing at a. Thus, 1/(2-a) is the break-even
point. As a varies through its possible range from to 1 , the break-
even point varies f rom V2 to 1. In order to provide a reasonably
gentle transition into the smallest spacing, it is preferable to have
a large region for the progression in spacing to occur. The largest
possible region would have a length of V2 and would occur when a van-
ishes. This, however, would leave no' room for a smooth transition
from endpoint spacing to the maximum spacing at a: a reasonably sized
region is needed here for the same reasons as in the situation with
the smallest spacing. Thus, a compromise is needed. As an example,
we consider the case with a = •=■. The transition into large spacing
then occurs over a third of the length while the final compression
after the break-even point occurs over the last 40? of the length.
The corresponding function is then

2x for < X < i

f(x) = ^ ^(3-5x) for i < X ^ 1 ^ (1|)

otherwise

542

where for notational convenience, we have dropped the subscript of
■r- which would have been required from the specialization of Eq. 3.

With the function for a = - , a weight for the forward mapping is
given by

t-t_ t -t
w = 1 + Mf[r^) + f^r:^-^^ (5).

for t 5* t and by w = 1 - B for t = tQ. The special treatment of tg
is required to remove a discontinuity caused by a contribution of -1
from f on each side when otherwise only one nonzero value would ap-
pear. The coefficient B is a control on the intensity of clustering.
In a direct sense, the spacing at the center is scaled by 1-3 to pro-
duce a smallest spacing in s. This spacing comes from Eq. 2 which
gives (1-3)dt at tg. For an n-point grid, it becomes
(1-3)(t -t . )/(n-1). As B varies from to 1 , the minimum spacing
varies from the original spacing down to 0. To avoid singularities,
we do not go down to but rather are interested in coming arbitrarily
close to 0. Unlike the earlier backward mapping, there is no price
for this arbitrary level of clustering intensity. This occurs because
the compensating areas for grid expansion are now in the positive di-
rection where there is no limit on size as there previously was when
the axis itself was being approached.

By use of the weight of Eq. 5 in Eq. 2, we obtain the forward
mapping

s = t . B{[tQ-tJg(^) . (tQ-tJg[i^)} (6a)

543

where

2 1 "^

x^ for ^ X < tIt

g(x) = -< l(1-x)(5x-1) for i < X < 1 ^ (5^)

otherwise

is the integral of f for increasing x. The interval lengths multiply-
ing each application of g result from a change of variable in each
corresponding integral. Geometrically, g appears as a simple bump
which leaves the axis (g(0) = 0) with zero slope (g'(0) = 0), monoton-
ically increases in the positive direction to reach a maximum, and
then monotonically descends back to the axis (g(1) = 0) to enter with
a negative slope (g'(1) = - 1), When assembled in the transformation,
6 scales a combination of positive and negative bumps which are joined
together with matching nonzero slopes. The addition to the line s = t
then represents a local distortion of it which causes clustering but
which preserves the uniform spacing elsewhere. An illustration of the
transformation is given in Fig. ^.

From a geometric viewpoint, we have simply taken the uniform
transformation s = t and have given it a local clockwise twist about
tQ. The severity of the twist is controlled by the slope at tg and to
some extent by the location of the maximum displacement from s = t.
This location corresponds to the break-even point where the spacing
from the transformation matches the uniform spacing from s = t; or in

other words, where the two slopes match. In the case of Eq. 6, the
choice of a = ■=■ led to a maximum displacement at x = 0.6. With the
more general piecewise construction, it occurs at x = 1/(2-a) and

544

Figure H : An elementary parameter transformation
in the forward direction of new t going into old s

thus can be controlled with the choice of a. The analytical formula-
tion is obtained by repeating the development with the f from Eq. 3
in place of f. Moreover, the locations can be controlled separately
on each side of Iq by using corresponding distinct a, and Op. The
analytical formulation is only altered by replacing the two applica-
tions of g in Eq. 6a with the corresponding generalizations g and

g of Eq. 6b.
2

545

ALTERNATIVE FORMULATIONS

While further shape control over the forward transformation de-
picted in Fig. 4 can be exercised with more exotic piecewise construc-
tions, we shall instead examine some alternatives which offer less
shape control but which present attractive options because of their
simplicity in statement. In this regard, we first note that the prev-
ious piecewise constructions achieved a high degree of algebraic sim-
plicity at the expense of doing it in a number of successive defining
intervals.

As a first step, we shall consider formulations which reduce the
number of defining intervals. Returning to the unit interval on which
we established f and then g, we shall consider a replacement. Noting
that second- and first-order zeros for g are desired respectively at
and 1, we are brought to consider the simplest positive bump function

g^(x) = x^Cl-x) (Y)

which satisfies the conditions when a > 1 and which is defined by one
segment. The derivative

f^(a) = x°'~^[a - (ct+1)x] (8)

assumes the value of -1 at x=1 and is also seen to vanish at x =
a/(1+a), which by our previous observations is the break-even point
with uniform spacing. As earlier, the location can be adjusted with a
choice of a. In distinction, this a alters the complexity of the
equation by creating larger powers when we wish to push the break-even
point towards 1. By contrast, the original piecewise development re-

546

quired only a shift of juncture point for the same purpose. The ap-
plication of Eq. 7 to create a forward transformation is direct and is
accomplished by simply replacing the g in Eq. 6 with the g of Eq. 7.
This can be done for either one or both of the intervals about tg and
each can have a separate adjustable a. Because f (1) = - 1, the con-
trol over minimum spacing by g is exactly the same. Thus, while we
also retain a capability to separately control the locations of break-
even points, we have been able to reduce the number of defining inter-
vals: non-zero values now appear on two rather than four intervals.

In continuation, we seek a further reduction to a single interval
with non-zero values. To do this, we shall construct a function which
will directly produce symmetric clusters. Rather than considering the
unit interval, we will develop the function on the larger interval
from -1 to 1 . To start, we form a symmetric positive bump with the
function

h^(x) = (1-x)°'(Ux)°' (9)

which is attached to the axis with vanishing first derivatives when a

> 1 and which has a single maximum value of unity when x = 0. At this

stage, a monotonically decreasing function through the origin is

needed as a factor to produce a negative slope at the origin and to

split the bump into a positive bump before origin and a negative bump

after the origin. If u(x) is such a function, then the derivative of

uh at X = is just u'(0) since h (0) = 1 and h'(0) = 0. The sim-
ot a a

pleat such choice is to set u(x) = - x. The associated function is
then

547

g (x) = - x(1-x)"(Ux)°'
a

(10)

and satisfies the properties g (±1) = g'(±1) = g (0) = and g'(0) =

a a a a

- 1. For a cluster interval of length 2T about tQ, we take x =
(t-tQ)/T in Eq. 10 and obtain the transformation

t . BTg J^]

<

for tQ-T ^ t ^ t +T

otherwise

> (11)

by vertically scaling the resultant bump pair by gT and then adding it
to the uniform mapping s = t. By direct differentiation, we have the
weight function of unity everywhere except on the interval about to

where it assumes the form

w = 1 . B[(2a.1)(— 0] - 1][1 - (—2) r'

(12)

The evaluation at tg gives precisely the earlier clustering control B

which produces the shrinking factor of 1-B. The motivation to get the

same control came from g'(0) = - 1 and the chain rule contribution of

a

1 /T as a factor. The break-even points with uniform spacing are given
when w = 1 in the interval about tg and are Just tg + T//2a+1 . As a
increases, these points then symmetrically approach tg. The largest
possible distance is bounded by T/-/3 in correspondence with a = 1
which is the lower bound for a. Thus, a can be used to control the
distance of break-even points from t^ over an interval from to T//3.
To have at least one-third of the interval for clustering, this choice
must be for a between 1 and H. Undoubtedly, variations on this theme
could be executed both to produce larger regions for the final clus-

548

tering compression and to insert a desired amount of asymmetry.

Rather than pursue these variations, we shall inspect the further
possibility of asymptotic approximation with the desire to simply de-
fine an elementary clustering transformation in one global statement
without having to establish a particular clustering interval. From
this viewpoint, such intervals are implicitly defined when the asymp-
totic decay is essentially complete. The impreciseness here then
gives us only a fuzzy definition. By contrast, however, we shall see
that the earlier break-even points can be established precisely.

As in the last case, we multiply a positive bump function by the
monotonically decreasing function - x which passes through the origin.
To start, we consider the bump function (1+x^) and arrive at

g^(x) = - xd+x^)"" (13)

which decays when a > 1. The uniform transformation s = t is now al-
tered for local clustering about tp by setting

s = t + BTg^(^] Uh)

Once a decay rate a is chosen, the length scale T is used to appropri-
ately shrink or expand the region of primary influence. By differen-
tiation, the associated weight is given by

(2a-1)[-^) - 1
w = 1 + e r^T (15)

This reduces to w = l-g at the cluster center tg and thereby retains
the meaning of the previous intensity controls 3. The decay rate a

549

controls the location of break-even points which from Eq. 15 appear at

a distance of T//2a-1 on either side of tg. An increase in a simply
causes a shift towards Iq relative to the scaling T. At the other ex-
treme, as a approaches 1, the shift is away from tg and is bounded by
T. Altogether, adjustments in decay rate allow break-even points to
be located anywhere between and T units away from the center tg. At
the extreme of T, the effective clustering region is enlarged beyond
T. To keep it, say within T units of tg, a somewhat conservative
choice is needed.

In the same spirit, we may also repeat the asymptotic construc-
tion with notably different analytical formulas. For example, we may
decide that a better bump function would be given by the Gaussian form

2
~~CtX

e and would then get

-ax^

g^(x) = - xe (16)

in place of Eq. 13- This would correspondingly be used in Eq. 1 i| with
the same interpretations for T and would lead to the weight

w = 1 + ^[2a{^f - i]exp[- a[-^]'] (17)

with the same clustering intensity control g. The positive damping
rate a is a control over the location of the break-even points rela-
tive to T. These are located at a distance of T/i/2a on either side of

to-

THE APPLICATIONS SETTING

To describe the setting in which applications are to be perform-

550

ed, we take note both of the general topic of grid generation and of
the order of application. Grid generation arose as a topic of study-
in response to the need for numerical simulations of realistic physi-
cal systems. It has now been the subject of three general reviews [1]
- [3], three major conferences [M] - [6] and one textbook [7]. A fun-
damental part of grid generation is the determination of pointwise
distributions on curves. This occurs because curves are basic con-
structive elements in virtually any approach to grid generation. At
the very least, they represent boundaries of two-dimensional regions
and are typically used to create bounding surfaces for three-dimen-
sional regions. The pointwise distribution on them directly influ-
ences the regional grid regardless of the method employed to generate
that grid. The further redistribution of families of curves or sur-
faces within a regional grid is also a typical consequence of the re-
distribution of points along curves.

To accomplish the redistribution of points along curves in a pre-
cise manner, we have developed herein the elementary operation of cre-
ating a single local cluster about a point. The application of the
operation to a succession of points can be ordered in either of two
natural ways: the points are taken one at a time or they are done
simultaneously. In correspondence, we may view the first as most
ideally suited to an interactive graphical environment while the sec-
ond appears more attractive for an automatic approach.

In the interactive setting, we assume that someone is sitting at
a graphics terminal or workstation with the capability to view the
pointwise distribution on the curve and to locate or insert pointwise

551

data by means of a cursor. For simplicity, we will assume that the
cluster center and endpoints are taken from the existing grid points
on the curve rather than at intermediate locations which would then
necessitate an interpolation. With this assumption, the cursor is
used to identify those grid points according to their indices. Since
the grid on the curve is the result of mapping a uniform grid in a
parameter s and since the corresponding uniform spacing can be taken
as unity, the indices directly give the parametric distance that the
endpoints s_ and s^ are from the center Sq. If we take Sq = 0, then
- s_ and s^ are respectively the number of grid points below and above
the cluster center. In terms of our new uniform parameter t, this be-
comes t_ = s_, tQ = 0, and t^ = s^. Next, the desired fractional de-
crease in spacing l-g is chosen for the center. The forward mapping
from Eq. 6 (or any of the equivalent variants) is now applied within
the interval from t_ to t^ to produce a local cluster of points about
Sq = 0. Unlike the center point and the points outside this interval,
the clustering has caused points to fall generally between the old
uniformly spaced points in s. If the curve is given analytically in
terms of s, then the old mapping is Just an evaluation at those in be-
tween points. Otherwise, for each new position in s, we must find the
unit grid interval that contains it and then linearly interpolate the
old map from s to the curve to get the new grid point location on the
curve. In this process, there is no need to operate on the points
outside of the cluster interval since they remain fixed. Upon appli-
cation of such an elementary clustering operation, the new distribu-
tion is viewed and then a judgment to stop or continue is made. If

552

the previous distribution is stored, then there is also the option to
easily restore it should we not like the result. Altogether, by ap-
plying the elementary clustering operations one at a time, we are able
to interactively manipulate the pointwise distributions on curves.

In the context where the judgments for clustering are determined
automatically for a collection of locations, it is more attractive to
perform a single mapping rather than a succession of mappings. Cer-
tainly, as the cluster regions overlap each other, the successive map-
ping approach becomes more repetitious and less efficient. To obtain
a single mapping, we may proceed from either of two viewpoints. The
first is to consider what would have occurred had we done successive
mappings. For any given order of mappings, the single mapping would
be a successive composition in the same order. By applying the chain
rule at each stage, the weight for the single mapping is just the pro-
duct of the weights from the elementary cluster maps. We note that
the elementary clustering weights are of the form w^^ = 1 + g.C. for

cluster functions C^ and intensities 3. where i = 1, 2 , n and n

is the number of clusters. The weight for the single mapping is then

w = (l + B^cjtl + B2C2] •••• (1 + B^C^) (18)

which is independent of the order of application. Unfortunately, the
product is not particularly convenient to integrate. As a conse-
quence, the linear B. -approximation

w = 1 + B.C. + B„C- + + B„C^ (19)

11 2 2 n n

is preferred and is also order-independent. Thus, the forward mapping

553

results from Eq. 2 by adding to uniform t = s the scaled bump pairs

from each B.C. as i = 1, , n. This superposition can then be

viewed in the format of Fig. ^1 where now the single twist about Sg =
tQ is replaced by n of them. In this context of n simultaneous clus-
ters, we note that a choice of specific intervals for each results in
a detailed partition of s that can be avoided if we employ asymptotic
approximations of the nature discussed in the section on alternative
formulations.

CONCLUSIONS

The capability to locally manipulate pointwise distributions on
curves was established through the introduction of an elementary oper-
ation for locally clustering points about any single point. The oper-
ation was created as a reparametrization where the spacing between new
and old parameters is prescribed by means of a weight function. Vari-
ous constraints upon the weights were established and the correspond-
ing transformations were examined. It was found that the forward
transformations from new to old are better because the composition of
mappings is simpler and because the clustering intensity control is
not limited as it is in the backward case.

The basic elements of construction were done in the most flexible
manner by using piecewise linear weights. This gave piecewise quad-
ratic transformations that were nontrivially defined over four inter-
vals, and more importantly, gave the fundamental guidelines for more
arbitrary constructions. Rather than pursue the greater degree of
shape control that is available from general piecewise polynomial con-

554

structions, alternatives were presented to reduce the number of inter-
vals of definition and thereby simplify the statement of the trans-
formations. This viewpoint was taken up to the stage where endpoints
of the local cluster region were only defined in a fuzzy sense by us-
ing asymptotic forms. These are attractive due to their simple global
expression in one statement rather than in the previous piecemeal
fashion. In summary, we first established a class of transformations
that are suitable for elemenatry clustering operations and then we ex-
plored a broad range of attractive candidates from that class.

The most obvious demand for the local redistribution of points
along curves occurs within the topic of grid generation and to some
extent provides a general applications setting. In a more particular
sense, the applications are considered to occur in sequence or simul-
taneously. Cases where only certain parts are simultaneous can be
subdivided into either of these two possibilities. The sequential
order of application is ideally suited to interactive graphics while
the simultaneous application is well suited to automation.

555

REFERENCES

[1] P.R. EISEMAN, "Grid generation for fluid mechanics computations,"
Annual Review of Fluid Mechanics , Vol. 17, 1985, pp. 487-522.

[2] J.F. THOMPSON, "Grid generation techniques in computational fluid
dynamics," AIAA Journal , Vol. 22, No. 11, 198^1, pp. 1505-1523.

C3] J.F. THOMPSON, Z.U.A. WARSI and C.W. MASTIN, "Boundary-fitted co-
ordinate systems for numerical solution of partial differential
equations - a review," Journal of Computational Physics , Vol. •47,
No. 1, 1982, pp. 1-108.

[4] K.N. GHIA and U. GHIA, Eds., Advances in Grid Generation , FED-

Vol. 5, American Society of Mechanical Engineers, New York, 1983.

[5] J.F. THOMPSON, Ed., Numerical Grid Generation , North-Holland, New
York, 1982. ~

[6] R.E. SMITH, Ed., "Numerical grid generation techniques," NASA CP
2166, 1980.

[7] J.F. THOMPSON, Z.U.A. WARSI and C.W. MASTIN, Numerical Grid Gen-

eration; Foundations and Applications , North-Holland, New York,
1985.

556

ON SIMILARITY SOLUTIONS OF A BOUNDARY LAYER PROBLEM
WITH AN UPSTREAM MOVING WALL

M. Y. Hussaini
Institute for Computer Applications in Science and Engineering

W. D. Lakin

Old Dominion Univeristy

and

Institute for Computer Applications in Science and Engineering

A. Nachman
Air Force Office of Scientific Research

ABSTRACT

This work deals with the problem of a boundary layer on a flat plate
which has a constant velocity opposite in direction to that of the uniform
mainstream. It has previously been shown that the solution of this boundary
value problem is crucially dependent on the parameter which is the ratio of
the velocity of the plate to the velocity of the free stream. In particular,
it was proved that a solution exists only if this parameter does not exceed a
certain critical value, and numerical evidence was adduced to show that this
solution is nonunique. Using Crocco formulation the present work proves this
nonuniqueness. Also considered are the analyticity of solutions and the
derivation of upper bounds on the critical value of wall velocity parameter.

Abbreviated title: Boundary layer on an upstream moving wall

Key words: non-uniqueness, Blasius equation, similarity solution

AMS classifications: 34B15 (Nonlinear boundary value problems)

76D10 (Boundary layer theory)

Research for the first and second authors was supported by the National
Aeronautics and Space Administration under NASA Contract Nos. NASl-17070 and
NASl-18107 while they were in residence at ICASE, NASA Langley Research
Center, Hampton, VA 23665-5225. ^^j

I . Introduction

The boundary layer on the upstream-moving flat plate at zero incidence
admits of the classical similarity transformation which reduces the relevant
partial differential equations to the Blasius equation.

f"-

' + ff" =

f(0) =

f'(0) = -X,

f'C") = 1,

X >

where f = i|)(x,y)//(2vx) , i); being the dimensional stream function, and v

the kinematic viscosity, and n = y//(2vx). This equation can be readily
integrated once to yield

f"(n) = f"(0)exp

/ f(z)dz

i.e. ,

f"(n) = f"(0)exp

y Xn^ -y / (n - z)^ f"(z)dz

using integration by parts twice. Obviously, the shear stress f"(n) has the
same sign as the skin-friction at the wall, f"(0). For X = 0, Weyl proved
the existence and uniqueness using function-theoretical methods. For X < 0,
Callegari and Friedman and Callegari and Nachman found it expedient to work
with the Crocco formulation, that is, in terms of shear stress g(=f") as the
dependent variable and tangential velocity u(=f'') as the independent
variable:

558

g(u)g"(u) + u = 0, -X < u < 1,
g'(-X) =
g(l) = 0.

For X £ 0, they proved existence, uniqueness, and analyticity of
solutions to Eq. (2) using an analytical function theory approach. For the
case X > 0, Hussaini and Lakin proved that a solution exists only for X
less than a critical value X . Their numerical results showed nonuniqueness
for X £ X , and the numerical value of X was found to be 0.3541... . In
this work, the nonuniqueness is established rigorously. Also, proof of
analyticity, and absolute monotonicity etc., is given. Certain analytical
upper bounds on X are established.

For convenience, we use the transformation x = u + X to map the
interval -X<u<l, to 0<x<l+X. So we consider the equations

g(x)g"(x) +(x-X)=0, 0<x<l+X (1.1)

(1.2)

g'(0) =
g(l + X) = 0.

2. Analiticity of Solutions

In this section, the following basic result will be proved:

THEOREM 1: There is a range of positive values of X such that the
positive continuous solution g(x) of the boundary value problem (1.1) and
(1.2) is analytic on the closed intervel [0,1 + X].

559

This theorem will be proved by considering a sequence of lemmas. The first
lemma required is :

1«KMMA. 1: The derivative g'(x) vanishes at one and only one point on
the interval < x < 1 + X. Further , g(x) has its maximum value at this
point .

Proof of Lemma 1 ; Equation (1.1) can be integrated using the initial
condition g'(0) =0 to give

^''"'■f iTTr''«' <2-i>

Thus, as the initial value a = g(0) > 0, both g(x) and g'(x) are positive
for < X <^ X. Also,

g"(x) = (X - x)/g(x) (2.2)

is positive for £ x < X and g"(X) = 0. The continuous solution g(x)
remains positive for X < x < 1 + X, and hence g"(x) is now negative. This
gives that g'(x) is a monotone decreasing function for x > X. As
g'(l + X) = -", there must thus be at least one point on the interval
(X,l + X) at which g'(x) vanishes. In fact, assuming that g'(x) vanishes
at more than one point leads to a contradiction, for suppose that g' vanishes
at both xj and X2 with xj < X2. Then, g" would have to vanish at least
once between these two points which is impossible as g" < for x > X. The
proof of Lemma 1 is concluded by noting that g"{-x.^) < implies that
g(xj^) must be the maximum value of g(x).

560

LEMMA. 2: The solution g(x) has a convergent power series expansion on
the closed Interval [x ,1 + X].

Proof of Lemma 2 : As g(x) is positive and dlf ferentlable for
X. <^ X < 1 + X, equation (2.2) shows that g(x) has derivatives of all
orders on this Interval. Further, expressions for these derivatives may be
obtained directly from the differential equation (1,1). Induction shows that
for n > 1, derivatives of g(x) satisfy the recursion relation

g("«) = - i j(„.„g- g("«) . 1 1 [(„!:!,) + iV) s<"' -<-''"^'

'n-k+3^ ^ k

g.n-..., (2.3)

where g^^ is the k-th derivative of g with respect to x and (^) is
the usual combinatorial symbol.

Let g(x ) = B, and consider the auxilliary function G(x) defined by

G(x) = B - g(x). (2.4)

Then, as 3 is the maximum value of g(x) , G(x) is non-negative for

X < X < 1 + X. Also, for all n ^ 1» G'^(x) = -g^^^x). Consequently,

equation (2.1) shows that G'(x) is positive on the interval x £ x < 1 + X,

From (1.1),

G"(x) = ijAr^ and G'"(x) = ^ '*' f ,^"
g(x) g(x)

are also both positive on this interval. The recursion relation (2.3) thus
shows that all derivatives of G(x) are non-negative on the closed interval
[x.,1 + X - e] where 1+X-x. >e>0. Hence, G(x) is absolutely

561

monotonic on this closed interval. A theorem of Bernstein [4] now gives
that G(x) has a convergent Taylor series expansion about the point x,
whose radius of convergence is not less than 1 + X - x . From the definition
of G(x), it immediately follows that for |x - x, | < 1 + X - x, , g(x) has
the convergent expansion

g(x) = I -^ (X - X )^ (2.5)

n=0

Application of a Tauberian theorem [5] further shows that the power series
(2.5) converges at the singular point x = 1 + X to the value g(l + X) =
completing the proof of Lemma 2.

To establish Theorem 1, it must be shown that for a nontrivial range of
positive values of X, the power series (2.5) for the solution g(x) of the
boundary value problem (1.1) and (1.2) converges at the left boundary point
X = 0. This will be accomplished in Lemma 3. A consequence of this
convergence will be an expansion for the initial value of g(x) as the series

, , X n n

CO { — 1 ) X

« = 6 + I -r-^g^"\x ). (2.6)

n=2 ^

LEMMA 3: There exists a positive value T such that if < X < T
then x^ < (1 + X)/2.

Lemma 3 gives that the left-hand boundary point x = lies inside the radius
of convergence of the power series expansion (2.5). Consequently, the
corresponding solution of the boundary value problem will be analytic. It
should be noted that the upper bound on x^ given in Lemma 3 is a sufficient,
but not a necessary, condition for convergence.

562

Proof of Lemma 3 ; Equation (1.1) may be integrated from to x using
the identity gg" = (gg')' - (g')^ and the initial condition g'(0) = 0. A
second integration from to Xi now gives the result

x2(x - 3X) 2 _ 2 ''l

-^-A ° ^ ^ + / (x. - Og-'^COd?. (2.7)

An upper bound on the right-hand side of (2.7) and a lower bound on the
maximum point x, are now required to establish the lemma.

A lower bound on x, may be obtained by using (2.1) and the fact that
g'(xj) = to obtain

As g(x) is monotone increasing on [0,x ], g(x) < g(.\) on [0,X], but
g(X) £ g(x) on [X,x ]. Equation (2.8) now gives

Xj > 2X. (2.9)

As g(x) has its only maximum at Xj by Lemma 1, an immediate lower
bound on g(x2) = 3 is 3 > a. A sharper lower bound on 3 can be obtained
from the expression

'■"^i g(S) ■<? - - -^ / ^^g^"? ".10)

obtained by integrating (2.1) from to x^. As g(x) £ 3, and by (2.9),
X. - X >^ X, equation (2.10) now gives the quadratic inequality

563

which Implies

3
6^ - aB - ±j-> (2.11)

3>^jV7T^^ (2.12)

2 2
A lower bound on B - a which follows from (2.12) is thus

3
5^ - a^ > ^ . (2.13)

Consider next bounds on the initial value a. Let X = 1 + X. Then,
integrating (2.1) from to X and using g(X) = gives

This relation may be rewritten in terms of strictly positive integrals as

which shows

„</''l2L:i|K?_iJ0 d5. (2.16)

The convexity of g(x) on [X,X] implies that on this interval

g(x) >^ g (X)»(X - x). Equation (2.16) now gives that a < (2g(X))~-^. As

a < g(X), this further implies

a^ O/z- C2.17)

564

Equation (2.15) does not lend itself to the derivation of a lower bound

2
on a . However, in the present consideration of analyticity, the required

bound can be obtained from a relation between a and 3 which follows from

the existence proof of Hussaini and Lakin [3]. This proof shows that if X

is positive and does not exceed a critical value, there is at least one

initial value a such that a positive continuous solution of the initial

value problem consisting of (1.1) and the conditions g(0) = a and g'(0) =

exists and has g(X) = 0, i.e., it is a solution of the boundary value

problem. Further, the solution of the initial value problem will be unique if

6 < 2a. (2.18)

It must be noted that a unique solution of the initial value problem dn°R not
imply a unique solution of the boundary value problem. This will be shown in

section 4.

2
A lower bound on a follows by using (2.18) in (2.12). The result is

2 X^
a > -^ . (2.19)

The final bound needed for use in equation (2.7) is an upper bound for
g'(x) on the interval [O.x^]. From (2.2), g"(x) is a monotone decreasing
function on this interval. Further, g"(X) = while the third derivative
of g is negative when x = X. Thus, g'(x) has its maximum value at x = X.
This implies that on [O.Xj]

< g'(x) < g'(X) = / A^dS. (2.20)

565

As g(x) >^ a on [0,X], equation (2.20) gives

< g'(x) <^ . (2.21)

An upper bound on the integral in equation (2.7) is thus

/ (x, - 5)g'^(5)d5 < I XxJ. (2.22)

o 1

Use of (2.13) and (2.22) in equation (2.7) implies

xj(x^ - ^. \) + 2X^ < 0. (2.23)

This relation gives that Xj will be less than X/2 for X in the range
< X < X" = 0.1176. The sufficient condition for analyticity is thus
satisfied for a range of positive values of X establishing Lemma 3 and
Theorem 1 .

Equation (2.9) implies that x^ cannot be less than X/2 if X > 1/3.
Indeed, direct numerical solution of the boundary value problem shows that x,
< X/2 when X < X = 0.32 and a lies on the upper branch in Figure 1. The
gap between the values of X and X is associated with fundemental problems
in obtaining sharper bounds on the initial value a. For example, equation
(2.15) implies

X
„ < r ^ (X - 0(g - X) r^ U - X) r^ (g - X)2 ,_ ., _,.

X x^ x^

566

Individually, the last two integrals in (2.24) are formally infinite, yet they

must cancel so as to give an order one upper bound. Direct numerical

2 2
calculations show that the upper bound on a is a < 0.219961. The upper

bound in (2.17) is thus conservative by over a factor of two.

It must again be noted that x^ < X/2 is only a sufficient condition for

analyticity. For values of a on the upper branch of Figure 1, solutions of

the boundary value problem can thus be expected to remain analytic for X

greater than X. Further insight can be gained by examining parameter values

for which the condition (2.18), which is sufficient for a unique solution of

the associated initial value problem, is maintained. Numerical results show

that (2.18) holds for all values of o on the upper branch of Figure 1. It

also holds for a on the lower branch of Figure 1 in the relatively small

range 0.351 < X < X and is violated over the remainder of the lower

branch. The behavior of 3 as a function of a is given in Figure 2. For

values of X associated with initial values on much of the lower branch of

Figure 1 , there must thus be serious doubts as to whether solutions of the

boundary value problem (1.1) and (1.2) are analytic.

3. An Upper Bound on X

The existence proof of Hussaini and Lakin [3] established the existence

of solutions of (1.1) and (1.2) for positive values of X less than a

critical value X . It was shown from (1.1) and (1.2) that X < 1/2. The
c c

value of X was also determined numerically in that work to be

X^ = 0.3541079... (3.1)

567

In this section, additional upper bounds for X will be obtained directly
from (1.1) and (1.2).

Using the identity that precedes equation (2.7), equation (2.1) can be
integrated from to x and the result integrated again from to X.
As g(X) = 0, this gives

^ ^^ : ^^^ = -^ + / (x - Og'^(OdC. (3.2)

6 2 Q

The right-hand side of (3.2) is intrinsically positive, and thus

X - 3X > 0. (3.3)

This relation immediately implies

X<V2. (3.4)

To obtain sharper bounds now requires the use of positive lower bounds

2
for a and the integral in (3.2). While no additional assumptions are

required to obtain (3.4), in what follows it will be necessary to assume that

3 < 2a. However, as noted previously, this condition is satisfied on the

entire upper branch in Figure 1. In particular, it is satisfied in the

limiting case when X = X .

Let the integral I(x) be defined by

^ 9

I(x) = / (X - ?)g'^(5)d5. (3.5)

Then, as I(X) > 0, equation (3.2) implies

568

X^(X - 3X) > 3a^. (3.6)

Replacing X by 1 + X and using (2.19) now gives the inequality

3X^ + 3X^ - 1 < (3.7)

which yields the improved bound

X < 0.47533. (3.8)

A slightly sharper bound can be obtained by noting that I(X) > I(X).
Let 6 = g(X). Then, g(x) < 5 on [0,X], so on this interval

2
g'^(x) > 2^. (2X - x)2. (3.9)

46^

This leads to the relation

I(X) >— ^ (5X + 16). (3.10)

An upper bound on 6 now follows from the fact that g(x) > a on [0,X] and

X ,. ,^2

r

6 = a + / ^^Jg^^ dC. (3.11)

In particular,

3
^2 < 2X_+_1. ^ (3^12)

Use of (3.12) in (3.10) then shows

569

Tfy^ s ^ (5X + 16)

I(X) > . (3.13)

60(2A + 1)

Equation (3.2) now gives

X^a - 3X) > 3a^ + 6I(X) (3.14)

which leads to the inequality

65X^ + 76X^ + lOX^ + 30X^ - 10 < 0. (3.15)

The solution of (3.15) is

X < 0.46824 (3.16)

which is only a marginal improvement over (3.8).

Even if the lower bound on l(x) is further sharpened by considering
this integral on the full interval [0,X], a significant decrease in the bound
on X is not obtained. Again, this is due to the difficulties associated
with obtaining sufficiently sharp bounds on the initial value a.

4. Non-uniqueness of Solutions for < X < X

c

Using direct numerical results, Hussaini and Lakin [3] have shown that if

X is positive and less than \^ then solutions of the boundary value

problem are not unique. For a fixed value of X in this range, as shown in

Figure 1 there are two initial values a which lead to solutions of the

boundary value problem. The purpose of this section is to prove this non-

570

uniqueness directly from (1.1) and (1.2). To this end, It Is convenient to
consider the normalized Initial value problem

hh" + t - L = 0, (A.l)

h(0) = 1, h'(0) = (4.2)

obtained from the initial value problem for g(x) by taking

g(x) = ah(t) with x = a^'^ t. (4.3)

The parameter L in (4.1) is related to a and X through the expression

L = a"^^^ X. (4.4)

If h(T) = and a(X) is given by

a = {(1 + X)/T}^/2, (4.5)

then g(X) = 0, so the solution of the initial value problem with initial
value (4.5) will also be a desired solution of the boundary value problem.
Equations (4.3) through (4.5) also imply that In terms of T and L

X-^ (4.6)

and

a = (T - L)~^/2. (4.7)

571

LEMMA 4: Let hj^(t) and h2(t) be solutions of the initial value
problem (4.1) and (4.2) corresponding to L values L, and L, ,
respectively. Then, if L2 > Lp h2(t) > hj(t).

Proof of Lemma 4 : For t « L, h(t) must be of the form 1 + Lt^/Z.

Thus, the lemma holds for small values of t. That it holds for < t < T

can now be shown by contradiction. Let "t be the first value of t at which

h^(t) = h2(t). As hj was previously less than h2, this requires
h^'(t) < h'j'(t"). But,

__ L, - t L, - t" L, - T

h^'(t) = -i— i— - > -1 h"(t). (4.8)

h2(t) h^(t) hj(t) ^

This contradiction establishes Lemma 4. Lemma 4 also shows that if
hj(Tp = and h2(T2) = 0, then h2(Tp > 0. This implies that:

COROLLARY: T2 > T^ .

The derivative h'(t) is given by an expression analogous to equation
(2.1). As h(0) is positive, both h(t) and h'(t) will be positive for
< t £ L. This shows that T > L. Consequently, the denominators in (4.6)
are strictly positive. The following lemma gives a sharper result:

LEMMA 5: T > 3L.

Proof of Lemma 5 : Equation (4.1) may be integrated twice from to
t using (4.2) to give

572

This implies

3 2 t
1/2 h^t) +^-IlH_ = l/2+ / (t - Oh'^COd?. (4.9)

h^(t) + -i- t^(t - 3L) > 0. (4.10)

Setting t = T and h(T) = now establishes the lemma.

Consider next the behavior of T as a function of L. It has already
been shown in Lemma 4 that T is a monotone increasing function of L.

LEMMA 6: T(L) is superlinear in L.

Proof of Lemma 6 : Let t, be the point at which h'(t,) = 0. As is the
case for the original initial value problem in the variable x, there is one
and only one such point, it lies in the interval L < t < T, and h(ti) is a
maximum value.

Equation (4.1) may be multiplied by h' and divided by h to give

hh" +lilil_J;l = 0. (4.11)

Integration from to t produces the result

1/2 h'^ + (t - L)lnh(t) - / lnh(5)d5 = 0. (4.12)

Evaluating (4.12) at tj^ now shows

573

^1

/ lnh(Od?
S = ^ -^ " lnh(tp ' <^-13)

Next, the expression

h(t) - 1 . /' '^ - L^^ - »

-Rcl ^^ '■'>■"'>

may be evaluated at t = L to give an expression for h(L).

'■™-^^i''T#''«-

(4.15)

As h'(t) is non-negative on the interval [0,L], h(t) is monotone
increasing, so h(t) <^ h(L). Use of this fact in (4.15) gives the quadratic

inequality

3

h^CL) - h(L) -3- > (4.16)

which implies h'^(L) 2 L /3. The solution h(t) has its maximum value at
t,. Consequently,

Mtp > /j- . (4.17)

One additional bound is needed before demonstrating the superlinear
behavior of T(L). The change of concavity of h(t) on the interval [0,t,]
due to the fact that h"(L) = precludes obtaining as a lower bound for h
on this interval the straight line which passes through the origin and the
point (tj,h(tj)), i.e., it cannot be shown that h(t) > h(t,)«t/t,. However,
for a given L, it is clear that h(t) can be bounded below on this interval
by a curve of the form

574

h(t.)t^
H(t;k) i (4.18)

^1

for a value of k > 1. As k increases, these curves become progressively
more convex. It should be noted that if H(t,k) provides a lower bound on
[0,tj] for the solution of (4.1) and (4.2) associated with L = L, then, by
Lemma 4, H(t,k) also provides a lower bound for solutions associated with
larger values of L.

This lower bound for h(t) on [0,ti] may be used to obtain an lower
bound for the integral in equation (4.13). In particular.

t

/ lnh(5)d? > t lnh(t ) - kt . (4.19)

111

Equation (4.13) now implies that

t^ >^ Inh(t^). (4.20)

Use of (4.17) then gives

^>^ >ki"(^)- (^-21)

The superlinear behavior of T(L) is thus established.

THEOREM 2: For positive values of X in the range < X < X ,
solutions of the boundary value problem (1.1) and (1.2) are not unique.

575

Proof of Therein 2 ; Consider the behavior of L as a function of X. By
(4.4), L(0) = 0. Equation (4.6) and the superlinear behavior of T with
repect to L shown in Lemma 6 now imply that the graph of X vs L must be
as in Figure 3. In particular, for a fixed positive X which is less than
X , there will be two distinct values of L. By the corollary to Lemma 4,
each value of L must correspond to a different value of T. Equation (4.5)
now shows that for the fixed value of X, two distinct values a. and a_
exist such that the solutions of the initial value problems with these a's
are solutions of the boundary value problem (1.1) and (1.2). Solutions of the
boundary value problem are thus not unique completing the proof of Theorem 2.

576

References

[1] A. J. Callegari and M. B. Friedman, "An analytical solution of a
nonlinear, singular boundary value problem in the theory of viscous
flows," J. Math. Anal. Appl ., 21 (1968), pp. 510-529.

[2] A. J. Callegari and A. Nachman, "Some singular, nonlinear differential
equations arising in boundary layer theory," J. Math. Anal. Appl ., 64
(1978), pp. 96-105.

[3] M. Y. Hussaini and W. D. Lakin, "Existence and nonuniqueness of
similarity solutions of a boundary-layer problem," Quart. J. Mech. Appl.
Math ., 39 (1986), in press.

[4] A. F. Tinman, Theory of Approximations of Functions of a Real Variable ,
Pergamon Press, England, 1963.

[5] N. Wiener, The Fourier Integral and Certain of its Applications , Dover,
New York, 1933.

577

A =0.3541078
c

^ 1— J.

.3 .4

Figure 1. Values of the parameter a = f"(0) for which f'(") = 1 as a
function of X .

578

1.0

0.8

/

/

/

/

0.6

0.4

/

/

0.2 -

/

0.1

0.2

0.3

0.4

0.5

OC

Figure 2. Values of the maximum value g of g(x) as a function of the
initial value g(0) = a. The dotted line is g = 2a .

579

Figure 3. The qualitative behavior of the parameter L in the initial

value problem (4.1) and (4.2) as a function of X.
580

ON THE ADVANTAGES OF THE VORTICITY-VELOCITY FORMULATION
OF THE EQUATIONS OF FLUID DYNAMICS

Charles G. Speziale

Institute for Computer Applications in Science and Engineering

NASA Langley Research Center, Hampton, VA 23665-5225

and
Georgia Institute of Technology, Atlanta, GA 30332

Abstract

The mathematical properties of the pressure-velocity and vorticity-
velocity formulations of the equations of viscous flow are compared. It is
shown that a vorticity-velocity formulation exists which has the interesting
property that non-inertial effects only enter into the problem through the
implementation of initial and boundary conditions. This valuable characteris-
tic, along with other advantages of the vorticity-velocity approach, are
discussed in detail.

Research was supported by the National Aeronautics and Space
Administration under NASA Contract No. NAS1-I8107 while the author was in
residence at ICASE, NASA Langley Research Center, Hampton, VA 23665-5225.

581

Two distinctly different approaches have been utilized in the literature
for the numerical solution of the equations of viscous flow in three-
dimensions. In the more common approach, the momentum equation, which
contains both the velocity and pressure, is solved numerically along with a
derived Poisson equation for the pressure (i.e., the pressure-velocity or
primitive variable formulation [1-3]). The alternative approach is based on
eliminating the pressure from the momentum equation by the application of the
curl. In this manner, a vorticity transport equation is solved numerically in
lieu of the momentum equation (i.e., the vorticity-velocity formulation [4-
6]). The purpose of the present note is to explore in more detail the
properties of these disparate numerical approaches. It will be shown that the
vorticity-velocity formulation has a striking advantage when applied to
problems in non-inertial frames of reference. More specifically, there exists
an intrinsic vorticity-velocity formulation wherein all non-inertial effects
(arising from both the rotation and translation of the frame of reference
relative to an inertial framing) only enter into the solution of the problem
through the implementation of initial and boundary conditions . This is in
stark contrast to the pressure - velocity formulation where non-inertial
effects appear directly in the momentum equation in the form of Coriolis and
Eulerian accelerations — a state of affairs which can give rise to a variety of
numerical problems [2]. A detailed exposition of this interesting property of
the vorticity-velocity formulation will be presented along with a brief
discussion of other advantages of this approach.

For simplicity, we will restrict our attention to the analysis of viscous
incompressible flow governed by the Navier-Stokes equation and continuity
equation which, respectively, take the form

582

3v 2

__+v«Vv=-Vp+vVv, (1)

V • V = 0, (2)

where v is the velocity vector, p is the pressure, and v is the kinematic
viscosity of the fluid. Here, the validity of (1) requires that the external
body forces be conservative and that the frame of reference be inertial. In
an arbitrary non-inertial frame of reference (which can rotate with a time-
dependent angular velocity f^(t) and translate with a time-dependent velocity
v^(t) relative to its origin 0), the Navier-Stokes equation takes the more
complex form [7]

8v , . 2

•r— +v« Vv + nxr+nx (fixr)+v_ + 2nxv = -Vp + vV v. (3)

Here, r is the position vector and the non-inertial terms on the left-hand
side of (3) are, respectively, referred to as the Eulerian, centrifugal,
translational, and Coriolis accelerations. The continuity equation still
assumes the same form (2) in any non-inertial framg of reference.

By the introduction of a modified pressure P which includes the
centrifugal and translational acceleration potentials, the non-inertial form
of the Navier-Stokes equation (3) can be simplified considerably. More
specifically, (3) can be written in the equivalent form

9v ,

_-+v« Vv + nxr+2nxv = -VP + vV v, (4)

where

583

P = P + Y (S • £)^ - 7 "^ ^^ + Vq . r. (5)

In the pressure-velocity formulation, equation (4) is solved in conjunction
with a Poisson equation for the pressure which is obtained by taking the
divergence of (4). Hence, the governing equations to be solved numerically in
this approach can be summarized as follows:

9v ^ 2

■-^+v*Vv + $^xr + 2nxv = -VP + vV v, (6)

V"^ P = - trfVv • Vv") + 2n • 0), (7)

subject to the initial and boundary conditions

V = V

'

at t = t„, (8)

1 = Xb

P = P^

on B. (9)

In (7) and (9), tr(«) denotes the trace, u is the vorticity vector, and B
denotes the boundary surface of the region. Of course, equations (6) and (7)
must be solved subject to the continuity equation (2). Since we are
considering general three-dimensional flow, a stream function solution does
not exist. Hence, the solution for the velocity v must be projected in some
suitable fashion onto the space of solenoidal vectors.

It is quite clear that the form of (6) and (7) (and, hence, their
mathematical character) change depending on whether or not the frame of

584

reference is inertial. Consequently, a particular numerical algorithm which
may be optimal for a given class of flows in an inertial frame of reference
may not be so for the same class of flows in a non-inertial framing. It will
now be demonstrated that the vorticity-velocity formulation does not suffer
from this deficiency.

The vorticity-velocity formulation is based on the vorticity transport
equation which is obtained by taking the curl of (4). This equation takes the

form

^~ 2

•5— + v • Vo) = 0) • Vv + vV 0) + 2fi • Vv - 2n (10)

in any non-inertial frame of reference where

0) = V X V (11)

is the vorticity vector. It is clear that the velocity and vorticity are also
connected through the Poisson equation

V^ v = - V X 0) (12)

which is a direct consequence of the vector identity

V X (V X v) = V(V • v) - V^ V. (13)

rsy r^

The intrinsic vorticity W, defined by

W = 0) + 2g, (14)

585

can be Introduced which represents the vorticity relative to an inertial frame
of reference. Since Q is spatially homogeneous (i.e., Vg = 0), it is a
simple matter to show that the non-inertial form of the vorticity-velocity
formulation can be written as follows:

aw 2

Jl + V • VW = W • Vv + vV W (15)

V^ V = - V X W. (16)

Equations (15) - (16) must be solved (in some region R with a boundary
surface B) subject to the initial and boundary conditions

W = (V X v)_ + 2fi-, at t = t- (17)

~ ~ ~ ~0

V = v_

on B. (18)

W = (V X v)„ + 2n

/v r^ e\j

Of course, it is well known that the vorticity, as well as the intrinsic
vorticity, are solenoidal, i.e.,

V • W = 0, (19)

(N/ /V

and, hence, the solutions for W and v must, in some suitable fashion, be
projected onto the space of solenoidal vectors.

This vorticity-velocity formulation of fluid dynamics represented by
equations (15) - (18) has the striking property that non-inertial effects only

586

enter into the solution of the problem through the Implementation of Initial
and boundary conditions . Consequently, the basic structure of the numerical
algorithm (I.e., the numerical formulation of (15) - (16)) will be independent
of whether or not the frame of reference is inertlal — a situation which
greatly enhances the general applicability of any Navler-Stokes computer code
which is developed based on this approach.

At this point, a few comments should be made concerning the alternate
ways in which the velocity field can be calculated in the vortlcity-velocity
formulation. Instead of solving the Poisson equation (16), it is possible to
solve the defining equation for vorticlty directly, i.e.,

Vxv = a) = W-2n, (20)

(see Gatski, Grosch, and Rose [6,8]). Of course, for plane or axisymmetrlc
flows, there exists a stream function ij; such that [7]

V = X X Vi|; (21)

tsj ts^ rsj '

V X (X X V\|)) = W - 2^, (22)

where ^ = Vx and x is the coordinate that the flow is independent of (for
plane flows, (22) reduces to the Poisson equation V i|) = W - 2n). While the
motion of the frame of reference does enter into the equations of motion In
these alternate vortlcity-velocity formulations, it does so in a much less
significant way than in the pressure-velocity formulation. To be specific,
the transport equation which is solved (i.e., equation (15)) does not contain

587

any frame-dependent terms and, at each time step, the partial differential
equation for the determination of the velocity field is only altered by the
addition of a constant forcing function in the form of 2n (the added term on
the right-hand side of (20) and (22)).

Finally, it would be of value to mention some other advantages of the
vorticity-velocity formulation. More difficulties have been known to arise in
the implementation of pressure boundary conditions than vorticity boundary
conditions [1,2] (of course, both boundary conditions must usually be
derived). Difficulties in satisfying the continuity equation in the pressure-
velocity formulation have also been known to give rise to numerical
instabilities [1]. Furthermore, in the vorticity-velocity approach, the
vorticity vector is calculated directly. This is of considerable value since
the vorticity field can play an important role in characterizing certain
features of turbulence [9]. While it is certainly not being suggested that
the pressure-velocity formulation be abandoned, this study does indicate that
the vorticity-velocity formulation can have distinct advantages when applied
to an important class of viscous flows.

Acknowledgment

The author would like to thank Dr. T. Gatski and Dr. M. Rose for some
valuable comments and criticisms of the original draft of this paper.

588

REFERENCES

[1] A. J. CHORIN, Math. Comp ., 22 (1968), 745.

[2] G. P. WILLIAMS, J. Fluid Mech ., 37 (1969), 727.

[3] D. A. ANDERSON, J. C. TANNEHILL, and R. H. FLETCHER, "Computational Fluid
Dynamics and Heat Transfer," McGraw-Hill, New York, 1984.

[4] S. C. R. DENNIS, D. B. INGHAM, and R. N. COOK, J. Comp. Phys ., 33 (1979),
325.

[5] H. F. FASEL, "Numerical Solution of the Complete Navier-Stokes Equations
for the Simulation of Unsteady Flows," Lecture Notes in Mathematics, No.
771, Springer-Verlag, Berlin, 1980.

[6] T. B. GATSKI, C. E. GROSCH, AND M. E. ROSE, to be published.

[7] G. K. BATCHELOR, "An Introduction to Fluid Dynamics," Cambridge
University Press, London, 1967.

[8] T. B. GATSKI, C. E. GROSCH, and M. E. ROSE, J. Comp. Phys ., 48 (1982), 1.

[9] E. LEVICH and A. TSINOBER, Phys. Letters, 93A (1983), 293.

"U.S. GOVERNMENT PRINTING OFFICE: 1986- 625-0l4:4000't 589

Standard Bibliographic Page

1. Report No. NASA CR-178076
ICASE Report No. 86-18

2. Government Accession No.

3. Recipient's Catalog No.

4. Title and Subtitle

ADVANCES IN NUMERICAL AND APPLIED MATHEMATICS

5. Report Date

March 1986

6. Performing Organization Code

7. Author(s)

J. C. South, Jr. and M. Y. Hussaini (editors)

8. Performing Orgeinization Report No.

86-18

9. Performing Organization Name and Address

Institute for Computer Applications in Science

and Engineering

Mail Stop 132C, NASA Langley Research Center
Hampton. VA 23665-5225

10. Work Unit No.

11. Contract or Gremt No.

NASI -17070; NASl-18107

12. Sponsoring Agency Nsime and Address

National Aeronautics and Space Administration
Washington, DC 20546

13. Type of Report cmd Period Covered

Contractor Report

14. Sponsoring Agency Code
50S-31-83-01

15. Supplementary Notes

Langley Technical Monitor;
J. C. South, Jr.

Final Report

To appear in Applied Numerical
Mathematics

16. Abstreict

This collection of papers covers some recent developments in numerical
analysis and computational fluid djmamics. Some of these studies are of a
fundamental nature. They address basic issues such as intermediate boundary
conditions for approximate factorization schemes, existence and uniqueness
of steady states for time-dependent problems, pitfalls of implicit time
stepping, etc. The other studies deal with modern numerical methods such as
total-variation-diminishing schemes, higher-order variants of vortex and
particle methods, spectral multidomain techniques, and front-tracking techniques.
There is also a paper on adaptive grids. The fluid dynamics papers treat the
classical problems of incompressible flows in helically-coiled pipes, vortex
breakdown, and transonic flows.

17. Key Words (Suggested by Authors(s))
Numerical Analysis
Computational Fluid Dynamics
Transonic Flows
Vortex Breakdown
Spectral Methods

18. Distribution Statement

34 - Fluid Mechanics & Heat Transfer
64 - Numerical Analysis

Unclassified - Unlimited

19. Security Classif.(of this report)
Unclassified

20. Security Classif.(of this page)
Unclassified

21. No. of Pages

597

22. Price
A25

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