STABILITY OF FINITE DIFFERENCE SCHEMES FOR HYPERBOLIC INITIAL - PDF document

STABILITY OF FINITE DIFFERENCE SCHEMES FOR HYPERBOLIC INITIAL BOUNDARY VALUE PROBLEMS Jean-Franc ¸ois Coulombel CNRS, Universit´ e Lille 1 and Team Project SIMPAF of INRIA Lille - Nord Europe Laboratoire Paul Painlev´ e (UMR CNRS 8524), Bˆ atiment M2, Cit´ e Scientifique 59655 Villeneuve d’Ascq Cedex, France Abstract. The aim of these notes is to present some results on the stability of finite difference approximations of hyperbolic initial boundary value problems. We first recall some basic notions of stability for the discretized Cauchy problem in one space dimension. Special attention is paid to situations where stability of the finite difference scheme is characterized by the so-called von Neumann condition. This leads us to the important class of geometrically regular operators. After discussing the discretized Cauchy problem, we turn to the case of initial boundary value problems. We introduce the notion of strongly stable schemes for zero initial data. The first main result characterizes strong stability in terms of a solvability property and an energy estimate for the resolvent equation. This result shows that the so-called Uniform Kreiss-Lopatinskii Condition is a necessary condition for strong stability. The main result of these notes shows that the Uniform Kreiss-Lopatinskii Condition is also a sufficient condition for strong stability in the framework of geometrically regular operators. We illustrate our results on the Lax-Friedrichs and leap-frog schemes. We also extend a stability result by Goldberg and Tadmor for Dirichlet boundary con- ditions to our framework of geometrically regular operators. In the last section of these notes, we show how to incorporate non-zero initial data and prove semigroup estimates for the discretized initial boundary value problems. We conclude with some remarks on possible improvements and open problems. These notes have been prepared for a course taught by the author in Trieste during a trimester devoted to “Nonlinear Hyperbolic PDEs, Dispersive and Transport Equations” (SISSA, May-July 2011). The material in the notes covers three articles, one of which is a collaboration with A. Gloria (INRIA Lille, France). These notes are also the opportunity to include some simplified proofs of known results and to give some detailed examples, which may help in clarifying/demystifying the theory. The author warmly thanks the organizers as well as the participants of the trimester for inviting him to deliver these lectures and for the very nice and stimulating atmosphere in SISSA. 1991 Mathematics Subject Classification. Primary: 65M12; Secondary: 65M06, 35L50. Key words and phrases. Hyperbolic systems, boundary value problems, finite difference schemes, stability. Research of the author was supported by the Agence Nationale de la Recherche, contract ANR-08-JCJC-0132-01. 1

2 JEAN-FRANC ¸ OIS COULOMBEL 1. Introduction 1.1. What is and what is not inside these notes ? These notes review the results derived in [4, 5, 6] on the stability of finite difference approximations for hyperbolic initial boundary value problems. In order to keep the length of the notes reasonable, the analogous results for hyperbolic partial differential equations, which have sometimes been proved quite some time ago, will be re- called without proof. This is mainly done to save space and to avoid introducing further notation. One crucial point in the analysis below is to understand why the techniques developed for partial differential equations are unfortunately not sufficient to handle finite difference schemes. Special attention is therefore paid to the main new phenomena that appear when considering discretized equations. Some examples are scattered throughout the text in order to explain how the general the- ory, which may look sometimes rather complicated, is often much simplified when one faces a specific example. In particular, the Lax-Friedrichs scheme, which is more or less the easiest discretization of a hyperbolic equation, serves as a guideline throughout Sections 2, 3 and 5. The notes are essentially self-contained. Of course, some familiarity with hyperbolic equations can do no harm, but the only basic requirements to follow the proofs are a good knowledge of matrices, some tools from real and complex analysis and a little bit of functional analysis. As far as hyperbolic boundary value problems are concerned, the reader might want to get first familiar with the theory for partial differential equations before reading the discrete counterpart that is detailed here. In this case, the books [3, chapter 7] or [2, chapters 3-5] are convenient references. However, the theory for finite difference schemes can also be seen as a first step towards the theory for partial differential equations since, as detailed below, some parts of the analysis are actually simpler in the discrete case. Even though the original results were not proved historically in this way, discrete problems can also be a constructive approximation method to obtain solutions of partial differential equations. (To be completely honest, the author is not fully convinced that this would be the most direct way to construct solutions of hyperbolic initial boundary value problems.) As far as numerical approximations are concerned, a convenient reference for our purpose is [8, chapters 5, 6, 11 and 13] where stability issues are analyzed, in particular for the discrete Cauchy problem. The techniques developed below are restricted to linear schemes for linear equations. Consequently, no knowledge of flux limiters, ENO/WENO schemes nor any other nonlinear high order approximation procedure is assumed. Extending some of the results below to such numerical schemes is definitely an open and challenging issue. 1.2. Some notation. Throughout these notes, the following notation is used: U := { ζ ∈ C , | ζ | > 1 } , U := { ζ ∈ C , | ζ | ≥ 1 } , S 1 := { ζ ∈ C , | ζ | = 1 } . D := { ζ ∈ C , | ζ | < 1 } , We let M d,D ( K ) denote the set of d × D matrices with entries in K = R or C , and we use the notation M D ( K ) when d = D . If M ∈ M D ( C ), sp( M ) denotes the spectrum of M , ρ ( M ) denotes the spectral radius of M , while M ∗ denotes the conjugate transpose of M . The matrix ( M + M ∗ ) / 2 is called the real part of M and is denoted Re ( M ). The real vector space of Hermitian matrices of size D is denoted H D . The vector space of real symmetric matrices of size D is denoted S D . If H 1 , H 2 ∈ M D ( C ) are two hermitian matrices, we write H 1 ≥ H 2 if for all x ∈ C D we have x ∗ ( H 1 − H 2 ) x ≥ 0. We let I denote the identity matrix, without mentioning the dimension. The norm of a vector x ∈ C D is | x | := ( x ∗ x ) 1 / 2 . The corresponding norm for matrices in M D ( C ) is also denoted |·| . We let ℓ 2 denote the set of square integrable sequences, without mentioning the indeces of the sequences (sequences may be valued in C d for some integer d ). The notation diag ( M 1 , . . . , M p ) is used to denote the diagonal matrix whose entries are (in this order) M 1 , . . . , M p . If the M j ’s are matrices themselves, then the same notation is used to denote the corresponding block diagonal matrix. The notation x 1 = x 2 ≤ x 3 = x 4 means that x 1 equals x 2 , x 3 equals x 4 , and x 2 is not larger than x 3 (and consequently, of course, x 1 is not larger than x 4 ). The letter C denotes a constant that may vary from line to line or even within the same line. The dependence of the constants on the various parameters is made precise throughout the text.