Discrete Shock Profiles: Existence and Stability

Abstract

Partial differential equations are often approximated by finite difference schemes. The consistency and stability of a given scheme are usually studied through a linearization along elementary solutions, for instance constants. So long as time-dependent problems are concerned, one may also ask for the behaviour of schemes about traveling waves. A rather complete study was made by Chow & al. [12] in the context of fronts in reaction-diffusion equations, for instance KPP equation; see also the monograph by Fiedler & Scheurle [17] for different aspects of the same problem. We address here similar questions in the context of hyperbolic systems of conservation laws. Besides constants, traveling waves may be either linear waves, corresponding to a linear characteristic field, or simple discontinuities such as shock waves of various kinds: Lax shocks, under-compressive shocks, overcompressive ones, anti-Lax ones.