Stability of Finite Difference Schemes for Hyperbolic Initial Boundary Value Problems

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. 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 kind and stimulating atmosphere in SISSA. Special thanks are addressed to Fabio Ancona, Stefano Bianchini, Gianluca Crippa and Andrea Marson for all the nice moments spent during the author’s stay in Trieste. 1991 Mathematics Subject Classification. Primary: 65M12; Secondary: 65M06, 35L50.