Stability of finite difference schemes for hyperbolic initial boundary value problems
Résumé
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 first 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 and check strong stability for various types of boundary conditions. We also extend a stability result by Goldberg and Tadmor for Dirichlet boundary conditions. In the last section of these notes, we show how to incorporate nonzero initial data and prove semigroup estimates for the discretized initial boundary value problems. We conclude with some remarks on possible improvements and open problems.
Domaines
Mathématiques [math]
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