Simple stability criteria for difference approximations of hyperbolic initial-boundary value problems. III

Mathl. Comput.

Modelling Vol. 20, No. lO/ll, pp. 49-54, 1994 Copyright@1994 Elsevier Science Ltd Printed in Great Britain. All rights reserved 0895-7177/94 $7.00 + 0.00

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Simple Stability Criteria for Difference Approximations of Hyperbolic Initial-Boundary Value Problems. III MOSHE GOLDBERG Department of Mathematics Technion - Israel Institute of Technology Haifa 32000, Israel To Heinz-Otto Kreiss Abstract-This revisionof [l], solicited by the Editors of Mathematical and Computer Modelling, gives an updated account of the Goldberg-Tadmor efforts to interpret the celebrated GustafssonKreissSundstr6m stability theory in order to obtain simple stability criteria for finite difference approximations associated with the hyperbolic systems ut = Au, + Bu + f in the quarter plane 2 2 0, t 2 0. Keywords-Hyperbolic bility.

initial-boundary

value problems, Finite difference approximations,

Sta-

Consider the initial-boundary value problem

wx, t)

-=

A&(x,

at u(x, 0) = d(x), uyo,

t3X

t> = cuyo,

t)

+

Bu(x, t) + f(x, t),

x 2 0,

t) + g(t),

t L 0,

Pa)

2 L 0,

(lb)

t L 0,

PC)

where (la) is a first order hyperbolic system of partial differential equations, (lb) is a given initial value, and (lc) is the boundary condition at x = 0. Here, u(x, t) = (u(l)(x, t), . . . , ZL(~)(X,t))’ is the unknown vector (prime denoting the transpose), f(x, t) and u’(x) are given n-vectors, and A and B are fixed n x n matrices such that A is diagonal of the form

A=

(“d$),

A'>o,

A"
with A’ and A” of orders k x k and (n - k) x (n - k), respectively. Further, C is an (n - k) x k coupling matrix, g(t) is a given (n - k)-vector, and

u*=

(u(l), . . . ,Jk) >‘( # =(

u(k+l)

,...,7P

>’

(3)

is a partition of u(z, t) into its outflow and inflow components, corresponding to the partition of A in (2). Introducing a mesh size Ax > 0, At > 0, such that A = At/Ax is fixed, and using the notation vj(t) E v (jAx, t), we approximate the hyperbolic system (la) by a general, consistent difference

49

M. GOLDBERG

50

scheme of the form Q-ivj

(t + At) = g Q7vj (t - rat) 7=0

+ Atb#),

j = p,p + 1,. . . ,

(4 7

=

-1 ,..‘> m,

where the n x n coefficient matrices A,, are polynomials in XA and AtB, the n-vectors bj(t) depend on f (2, t) and its derivatives, and E is the translation operator defined by

Evj(t) = vj+l(t). In order for the scheme in (4) to have a unique solution, we augment it by prescribing initial values vj (rat)

= vj” (rat)

,

r=O,...,m,

j =0,1,2

,***,

(5)

and specifying at each time level t = rat, r = m, m + 1, . . . , boundary values vj (t + At), j = o,... ,p - 1. Such boundary values are most generally determined by solvable boundary conditions of the form

S?iVj(t +

At) =

2 S!j)vj (t -

TAt) + Atdj(t),

j=o,...,p-1,

r=O

(6)

where the n x n matrices C$$ depend on A, At B and C, and the n-vectors dj(t) are functions of f(z, t), g(t) and their derivatives. We shall always assume that the basic scheme in (4) is stable for the pure Cauchy problem associated with (la). It is well known, however, that the introduction of unsuitable boundary conditions may destroy the overall stability of approximation (4)-(6). Thus, we are concerned here with interpreting the Gustafsson-Kreiss-Sundstrom stability theory in [2] in order to obtain simple stability criteria for our approximation. While unable to meet this goal for general boundary conditions of type (6), we managed to obtain satisfactory results under the following mild restrictions: writing the matrix coefficients C$$, in harmony with the partition of A in (2), as

(7a)

we assume that when B = 0, then @(A =

0

$Uj,

= 0

for 0 < 0,

(7c)

thy@)

a,

the C$)

are diagonal,

@r(j) = 0 (r7

(7b)

independent of j, for u > 0 and for u = 0, r > 0.

(7d) (7e) (7f)

These constraints on the C$, particularly (7f), considerably relax the corresponding assumptions in [3,4]. Here, (7b,c,d) imply that for B = 0, the outflow boundary conditions are, as often expected, right-sided, independent of inflow values, and translatory (i.e., determined at all

51

Simple Stability Criteria for Difference Approximations

boundary

points

able outflow diagonal

by the same coefficients).

boundary

point

condition

(7e) is satisfied

by all reason-

since for B = 0, the C,,I(‘) often reduce to polynomials (7f) maintains that for B = 0, the inflow boundary conditions

conditions

block A’. Finally,

boundary

Furthermore,

may depend

only on inflow values located

to the left of the boundary

in the at each point

in

question. In a forthcoming paper, we shall discuss further the boundary conditions in (6)-(7), and show that the conclusions of the stability analysis in [3,4] hold verbatim for our approximation in (4)-(7). We thus proceed now to describe these conclusions, omitting the proofs. The first step in our stability analysis is to reduce (4)-(7) to a scalar homogeneous mation.

This is done by introducing

the outflow scalar hyperbolic

x 2 0, t 2 0, a = constant (which requires

no boundary

condition),

for which (4) reduces

approxi-

problem

to a scalar,

> 0 stable,

homogeneous

basic scheme of the form

Q-Ivj

(t +

At) =

2 Q+ vj (t - Tat),

j=p,p+l,...,

7=0

(94 -l,...,m,

7=

re d uces to scalar,

and (6)-(7)

homogeneous

translatory

conditions

S_lvj(t+At)=~S,vj(t_7at), r=o

of the form

j=o,...,p-1,

WI

S, = 2 c,, Em, o=o

7=

-l,...,m,

in a and X E At/Ax. where the coefficients aoT and c, T are polynomials Referring to (9) as the reduced upprodmation, we can prove the following

theorem.

THEOREM 1. [3,4]. Approximation (4)-(7) is stable (in the sense of [2]) if and only if the reduced approximation (9) is stable for every (positive) eigenvalue a of the diagonal block A’ in (2). That is, approximation (4) -( 7) is stable if and only if the k scalar outffow components of its principal part are all stable. This reduction theorem implies that from now on we may restrict our stability study to the reduced approximation (9). In order to introduce our stability criteria for the reduced approximation, we use the scalar coefficients of the reduced basic scheme (9a) to define the basic characteristic function

@(z,tc) =

2 0=-p

Similarly, using the coefficients characteristic function

au,-1 [

of the reduced

9(r, 6) = 2 &,-I o=o [

cm aor

*-7-l

boundary

conditions

- 2 c,, t-r-1 +=o

Now, putting Q(z, K) = I% we obtain

the following

theorem.

fp. 1

T=O

&)I + I%,

k)I ,

(9b), we define the boundary

1

KU.

M. GOLDBERG

52

THEOREM 2. [3,4]. The reduced approximation (9) is stable if: (a) The reduced scheme (9a) is stable for the pure Cauchy problem associated with (8). (b) We have either d@ d@ (104 Z * dd < O or R (z = -1, K = -1)

> 0;

(lob)

and in addition for all IzI = (~1 = 1, K # 1, (z, IC) # (-1,

fl (z,K) > 0, R (z, K = 1) > 0, 0 (5 K) > 0,

-l),

(1Oc)

for all ItI = 1, z # 1,

(104

for all IzI 2 1, 0 < 1~1< 1.

(1Oe)

The advantage of this formulation of Theorem 2 is clarified by the following lemma, in which we provide helpful sufficient conditions for each of the four inequalities in (lob+)

to hold.

LEMMA 1. [3,4].

(4 (b)

Inequalities (lOb,c) hold if either the reduced scheme (9a) or the reduced boundary conditions (9b) are dissipative. InequaJity (10d) holds if any of the following is satisfied: The reduced scheme is two-level (i.e., m = 0 in (9a)). The reduced scheme is three-level (m = 1 in (9a)), and Q(z = -1,

(cl

K = 1) > 0.

(11)

(iii) The reduced boundary conditions are at most two-level, and at least zero-order accurate as an approximation to equation (8). (iv) The reduced boundary conditions are three-level, at least zereorder accurate as an approximation to (8), and (11) is satisfkd. Inequality (1Oe) holds if the reduced boundary conditions fulfill the von Neumann condition, and are either explicit or satisfy

S-l(K)

E

2

cc,_1

KU # 0,

for all 0 < 1~1I 1.

Relating to the terminology in the above lemma and in Theorem 2, we recall for the sake of completeness the following definitions:

(4

The reduced scheme (9a) is dissipative if the roots Z(K) of the equation Q (z, K) = 0 satisfy

l+>I < 17 (b)

for all K with Ifi.1= 1, K # 1.

The reduced scheme (9a) fulfills the von Neumann Condition if the roots of @(z, IS) satisfy

IZ(~)l I 1,

Cc)The

for all K with 1~1= 1.

reduced scheme (9a) is stable for the Cauchy problem associated with (8), if it fulfills the von Neumann Condition; and if Z(K) is a root of Q(z,K) such that 1~1 = Iz(~)l = 1, then Z(K) is a simple root of @(z,K).

Simple Stability Criteria for Difference Approximations

53

Analogous definitions hold for the reduced boundary conditions in (9b), where the roots of a(.~, IC) are replaced by those of Q(z, K). Clearly, for the reduced scheme as well as for the reduced boundary conditions, dissipativity implies the von Neumann condition. Evidently, the stability criteria obtained in Theorem 2 depend both on the basic scheme (9a) and on the boundary conditions (9b), but not on the intricate and often complicated interaction between the two. Consequently, Theorem 2, aided by Lemma 1, provides, in many cases, a convenient alternative to the Gustafsson-Kreiss-Sundstrom stability criterion in [2]. We conclude this note with three typical examples. 1. Consider an arbitrary, stable reduced scheme, in (9b) be generated by either the explicit, first-order

and let the reduced accurate, right-sided

EXAMPLE

ditions

vj (t + At) = vj(t) + Xa[vj+r(t) or by its implicit

O
-v&)],

j=o

Aa>O,

(t + At) - vj (t + At)],

These two-level boundary conditions are dissipative (see 151, E xamples the von Neumann condition. Further, for (12) we have Re[S_r(K)] 1, therefore,

EXAMPLE 2.

ary conditions

p-l,

analogue,

~j (t + At) = ~j(t) + Xo[~j+r

By Lemma is stable.

,...)

boundary conEuler scheme,

= 1 + Xa[l

inequalities

(lobe)

- Re(K)] # 0,

j=o

,.‘.,

p-l.

(12)

3.5 and 3.6), hence, fulfill

14 I 1.

hold; so by Theorem

2, the reduced

approximation

Take an arbitrary, stable, two-level reduced scheme, and define the reduced by horizontal extrapolation of order e - 1: %

(t + At) =

2:

k=l

vj+k (r + At) ,

(-l)“+l

bound-

j=o,...,p-1.

0

Here,

qz,

k) = 1 - f:

(i)

(--l)k+‘/$

= (1 - ti)l;

k=l

so @(z, K) # 0 for K. # 1, which directly gives (lOb,c,e). Moreover, since the reduced scheme is two-level, Lemma 1 (b) (i) implies (10d) , and Theorem 2 again yields stability. This result may fail (e.g., [6]) for both dissipative and nondissipative schemes of more than two time levels. EXAMPLE 3.

Let the reduced

approximation

in (9) consist

uj (t + At) = Vj (t - At) + Xa[vj+l(t) with oblique

extrapolation

- vj_l(t)],

of the (stable) O
of order C - 1 at the boundary: v. (t + At) = &

(:)(-l)itruk

[t - (k - l)At].

k=l

We have @(Z, 6) = 1 - Z-2 - A a Z-r (& - /c-l) .

so

m

d@

dz *dlE

=$
z=n=_l

Also, for L = K # fl, and R(z,lc) Hence,

(lOa,c-e)

For further

I jQ(Z,K)I = 11 - .-l,le

hold, and by Theorem

examples,

see [3,5,6].

2, stability

> 0, follows.

for z # K.

Leap-Frog j=1,2,3

scheme, ,...,

M. GOLDBERG

54

REFERENCES 1.

M. Goldberg, Simple stability criteria for difference approximations of hyperbolic initial-boundary value problems, II, In Third International Conference on Hyperbolic Problems-Theory, Numerical Methods and Applications, (Edited by B. Engquist and B. Gustafsson), pp. 519-527, Studentliteratur, Lund, (1991). 2. B. Gustafsson, H.-O. Kreiss and A. SundstrGm, Stability theory of difference approximations for mixed initial boundary value problems, II, Math. Comp. 26, 649-686 (1972). 3. M. Goldberg and E. Tadmor, Convenient stability criteria for difference approximations of hyperbolic initial-boundary value problems, II, Math. Comp. 48, 503-520 (1987). 4. M. Goldberg and E. Tadmor, Simple stability criteria for difference approximations of hyperbolic initial-boundary value problems, In Nonlinear Hyperbolic Equations-Theory, Numerical Methods and Applications, (Edited by J. Ballmann and R. Jeltsch), pp. 179-185, Vieweg Verlag, Branschweig, (1989). 5. M. Goldberg and E. Tsdmor, Scheme-independent stability criteria for difference approximations of hyperbolic initial-boundary value problems, II, Math. Comp. 36, 605-626 (1981). 6. M. Goldberg and E. Tadmor, Convenient stability criteria for difference approximations of hyperbolic initial-boundary value problems, Math. Comp. 44, 361-377 (1985).