Stability of finite difference schemes for hyperbolic systems in two space dimensions

Jean - Franc¸Ois Coulombel

Description

Stability of finite difference schemes for hyperbolic systems in two space dimensions Jean-Franc¸ois Coulombel CNRS & Universite Lille 1, Laboratoire Paul Painleve, UMR CNRS 8524, Cite scientifique, 59655 VILLENEUVE D'ASCQ Cedex, France Email: April 6, 2005 Abstract We study the stability of some finite difference schemes for hyperbolic systems in two space dimensions. The grid is assumed to be cartesian, but the space steps in each direction are not necessarily equal. Our sufficient stability conditions are shown to be also necessary for one concrete example. We conclude with some numerical illustrations of our result. AMS subject classification: 65M12, 65M06, 35L45. Keywords: Hyperbolic systems, finite difference schemes, stability. 1 Introduction Finite difference schemes are commonly used to approximate the solutions to hyperbolic systems of conservation laws. In this paper, we are interested in the stability of such finite difference schemes when applied to constant coefficients hyperbolic systems in two space dimensions. When applied to variable coefficients or nonlinear systems, the Courant-Friedrichs-Lewy condition that we derive can be seen as a local condition that needs to be satisfied in each cell of the grid. We consider a symmetric hyperbolic system in two space dimensions: { ∂tu + A1 ∂x1u + A2 ∂x2u = 0 , t ≥ 0 , x ? R2 , u|t=0 = u0 , x ? R2 .

• lax-friedrichs


• ?2 sin


• sin2 ?2


• normal matrix


• finite difference schemes


• hyperbolic systems


• ?2


• when constructing


• ?1

Sujets

• Painlevé
• Normal matrix
• Finite difference method

Informations

• Publié par: profil-urra-2012
• Nombre de lectures: 40
• Langue: English

Extrait

Stability of nite di erence schemes for hyperbolic systems
in two space dimensions
Jean-Fran cois Coulombel
CNRS & Universite Lille 1, Laboratoire Paul Painleve, UMR CNRS 8524,
Cite scienti que, 59655 VILLENEUVE D’ASCQ Cedex, France
Email: [e-mail]
April 6, 2005
Abstract
We study the stability of some nite di erence schemes for hyperbolic systems in two
space dimensions. The grid is assumed to be cartesian, but the space steps in each direction
are not necessarily equal. Our su cien t stability conditions are shown to be also necessary
for one concrete example. We conclude with some numerical illustrations of our result.
AMS subject classi cation: 65M12, 65M06, 35L45.
Keywords: Hyperbolic systems, nite di erence schemes, stability.
1 Introduction
Finite di erence schemes are commonly used to approximate the solutions to hyperbolic systems
of conservation laws. In this paper, we are interested in the stability of such nite di erence
schemes when applied to constant coe cien ts hyperbolic systems in two space dimensions. When
applied to variable coe cien ts or nonlinear systems, the Courant-Friedrichs-Lewy condition that
we derive can be seen as a local condition that needs to be satis ed in each cell of the grid.
We consider a symmetric hyperbolic system in two space dimensions:
(
2@ u + A @ u + A @ u = 0; t 0; x2 R ;t 1 x 2 x1 2 (1)
2u = u ; x2 R :0jt=0
The matrices A , and A belong to M (R), and are symmetric, so that the Cauchy problem (1)1 2 d
2 2is well-posed in L (R ), see e.g. [3]. Moreover, the solution of (1) satis es
8t 0; ku(t)k 2 2 =ku k 2 2 : (2)0L (R ) L (R )
We introduce a nite di erence approximation of (1). Let x , and x denote some space1 2
nsteps in the x , and x directions, and let t denote the time step. Then the vector u , where1 2 j;k
(n;j;k)2 NZZ, denotes an approximation of u(nt;j x ;k x ). Following [1], we de ne1 2
t t
:= ; := :1 2x x1 2
We refer to [1, chapter IV.3], and [2, chapter 6] for a general description of nite di erence
schemes for two-dimensional hyperbolic systems, and we shall thus assume that the reader is
2familiar with the basic L stability theory of nite di erence schemes (see e.g. [1, page 348]).
In this paper, we shall study the stability of four nite di erence schemes:
1 The two-dimensional Lax-Friedrichs scheme:
1 1 2n+1 n n n n n n n nu = (u +u +u +u ) A (u u ) A (u u ):1 2j;k j 1;k j+1;k j;k 1 j;k+1 j+1;k j 1;k j;k+1 j;k 14 2 2
(3)
The dimensional-splitting Lax-Friedrichs scheme:
1 n+1=2 1n n n nu = (u + u ) A (u u );1j 1;k j+1;k j+1;k j 1;kj;k 2 2 (4)
1 n+1=2 n+1=2 2 n+1=2 n+1=2n+1u = (u + u ) A (u u ):2j;k j;k 1 j;k+1 j;k+1 j;k 12 2
The two-dimensional Godunov scheme:
1 1n+1 n n n n n nu = u A (u u ) jA j(2u u u )1 1j;k j;k j+1;k j 1;k j;k j+1;k j 1;k2 2
2 2n n n n nA (u u ) jA j(2u u u ): (5)2 2j;k+1 j;k 1 j;k j;k+1 j;k 12 2
The dimensional-splitting Godunov scheme:
n+1=2 1 1n n n n n nu = u A (u u ) jA j(2u u u );1 1j;k j;k j+1;k j 1;k j;k j+1;k j 1;k2 2 (6)
n+1=2 2 n+1=2 n+1=2 2 n+1=2 n+1=2 n+1=2n+1u = u A (u u ) jA j(2u u u ):2 2j;k j;k j;k+1 j;k 1 j;k j;k+1 j;k 12 2
We do not know whether the terminology is really standard, but we hope that it is clear enough.
Recall that in (5), and (6), the matricesjA j are de ned as follows: let P denote orthogonal1;2 1;2
matrices that diagonalize A :1;2
1 1P A P = diag ( ;:::; ); P A P = diag ( ;:::; ): (7)1 1 1 d 2 2 1 d1 2
Then the matrices jA j, and jA j, are given by:1 2
1 1P jA jP = diag (j j;:::;j j); P jA jP = diag (j j;:::;j j): (8)1 1 1 d 2 2 1 d1 2
Observe that jA j, and jA j are symmetric, nonnegative matrices. They are positive de nite if1 2
A , and A are nonsingular.1 2
When = , the stability of (3) was completely analyzed in [4], even in the case of variable1 2
coe cien ts. The extension to di eren t space steps is easy, but we give it here to enlight the
di erence between the stability criteria for (3) and (5).
In all what follows, the spectral radius of a square matrix M with complex entries is denoted
(M). Our main result is the following:
2 2Theorem 1. The scheme (3) is stable in ‘ (Z ) if
1
p8#2 [0;2]; ( cos#A + sin#A ) : (9)1 1 2 2
2
2 2 The scheme (4) is stable in ‘ (Z ) if, and only if
max( (A ); (A )) 1: (10)1 1 2 2
22 2 The scheme (5) is stable in ‘ (Z ) if
(A ) + (A ) 1: (11)1 1 2 2
If A , and A are nonsingular, and if (A ) + (A ) < 1, then the scheme (5) is1 2 1 1 2 2
dissipative (in Kreiss’ sense) of order 2. Namely, if G( ; ) denotes the symbol of the1 2
scheme (5), there exists a constant c > 0 such that

2 28( ; )2 ; ; ; (G( ; )) 1 c ( x ) + ( x ) :1 2 1 2 1 1 2 2
x x x x1 1 2 2
2 2 The scheme (6) is stable in ‘ (Z ) if, and only if
max( (A ); (A )) 1: (12)1 1 2 2
For the schemes (3), and (5), Theorem 1 only gives su cien t stability conditions. For
a particular system, one may hope to get less restrictive stability conditions. However, the
following result shows that the conditions of Theorem 1 are optimal in the general case (that is,
they can not be improved for all symmetric hyperbolic systems):
Theorem 2. Let A , and A be given by1 2

0 1 1 0
A = ; A = :1 21 0 0 1
Then we have the following necessary and su cient conditions:
p
2 2 The scheme (3) is stable in ‘ (Z ) if, and only if 2 max( ; ) 1, which is equivalent1 2
to (9).
2 2 The scheme (5) is stable in ‘ (Z ) if, and only if + 1.1 2
The paper is organized as follows. In section 2, we prove the rst two items of Theorem 1,
and we also give a Lax-Friedrichs type scheme that is always unstable. We give this example
in order to highlight the fact that one should be cautious when constructing two-dimensional
schemes by simply adding one-dimensional schemes in each direction. Such an operation may
yield instabilities. In section 3, we prove the last two items of Theorem 1. Then in section 4, we
prove Theorem 2. Eventually, in section 5, we compare the dissipativity of the Lax-Friedrichs
and Godunov schemes with the help of numerical simulations. We shall also discuss the choice
of the space steps.
2 Stability of Lax-Friedrichs type schemes
2.1 An unstable Lax-Friedrichs type scheme
There are many possible ways to construct a nite di erence schemes in two space dimensions.
As a rst guess, one could think that it is enough to add the one-dimensional Lax-Friedrichs
uxes in each direction. Such a procedure yields the following scheme (see e.g. [1, page 346]):
1 1 2n+1 n n n n n n n n nu = (u +u +u +u 2u ) A (u u ) A (u u ):1 2j 1;k j+1;k j;k 1 j;k+1 j;k j+1;k j 1;k j;k+1 j;k 1j;k 2 2 2
The symbol G of this scheme is computed by using a Fourier transform in the space variables.
We obtain:

G( ; ) = cos( x ) + cos( x ) 1 I i( sin( x )A + sin( x )A ) :1 2 1 1 2 2 d 1 1 1 1 2 2 2 2
In particular, when x = x = , the symbol G equals 3I , and the scheme is unstable1 1 2 2 d
2 2in ‘ (Z ).
32.2 Stability of Lax-Friedrichs scheme
We now study the scheme (3). Its symbol is computed by applying a Fourier transform in the
space variables. We get
1LFG ( ; ) = cos( x )+cos( x ) I i( sin( x )A + sin( x )A ) : (13)1 2 1 1 2 2 d 1 1 1 1 2 2 2 2
2
LFThe matrices A are symmetric. Therefore, the matrix G ( ; ) is normal for all ( ; ).1;2 1 2 1 2
The scheme (3) is thus stable if, and only if:
2 LF LF8( ; )2 R ; I G ( ; ) G ( ; ) 0:1 2 d 1 2 1 2
To simplify the computations, we denote = x , k = 1;2. Following [4], we computek k k

1 1LF LF 2 2 2I G ( ; ) G ( ; ) = (sin + sin ) + (cos cos ) Id 1 2 1 2 1 2 1 2 d2 4
2
sin A + sin A :1 1 1 2 2 2
Choosing # such that
q q
2 2 2 2sin = cos# sin + sin ; sin = sin# sin + sin ;1 1 2 2 1 2
we end up with

1 2LF LF 2 2I G ( ; ) G ( ; ) (sin + sin ) I cos#A + sin#A :d 1 2 1 2 1 2 d 1 1 2 2
2
The rst item of Theorem 1 follows, by recalling that for a hermitian matrix H (and more
generally for a normal matrix), the hermitian norm of H (that is, the norm induced by the
dhermitian norm in C ) equals the spectral radius (H).
2.3 Stability of the dimensional-splitting Lax-Friedrichs scheme
We now study the scheme (4). Its symbol is given by

LFsG ( ; ) = cos( x )I i sin( x )A cos( x )I i sin( x )A : (14)1 2 2 2 d 2 2 2 2 1 1 d 1 1 1 1
Choosing either = 0, or = 0, it is clear that the stability of (4) implies the stability of1 2
each corresponding one-dimensional Lax-Friedrichs schemes. Therefore, if (4) is stable, then
(A ), and (A ) are both less than 1.1 1 2 2
Assume now that both (A ), and (A ) are less than 1. From (14), we see that the1 1 2 2
LFssymbol G ( ; ) is the product of two normal matrices, each of which has a spectral radius1 2
bounded by 1. For a normal matrix, the spectral radius coincides with the hermitian norm,
LFswhich implies that the hermitian norm of G ( ; ) is less than 1. This ensures that (4) is1 2
stable.
3 Stability of Godunov type schemes
3.1y of the two-dimensional Godunov scheme
The symbol of the Godunov scheme (5) is

x x1 1 2 22 2G( ; ) = I 2 sin ( )jA j + sin ( )jA j1 2 d 1 1 2 2
2 2
i( sin( x )A + sin( x )A ) : (15)1 1 1 1 2 2 2 2
4In general, the matrix G( ; ) is not normal for all values of ( ; ). As a matter of fact, the1 2 1 2
reader can check that G( ; ) is normal if, and only if the matrices A , and A satisfy1 2 1 2
A jA j jA jA = A