Numerical Investigation of Examples of Unstable Viscous Shock Waves

Abstract

Recently there has been a growing interest in stability problems for viscous strong shock wave solutions of systems of conservation laws. In this paper we numerically investigate a family of unstable viscous shock waves the existence of which was proved in [2]. This example occurs in a perturbed version of the rotationally invariant cubic model, a non-strictly hyperbolic 2 x 2 system of conservation laws. By this procedure we obtain an exponentially unstable shock wave of Lax type. We verify the instability by computing the unstable eigenvalue of the operator linearized at the traveling wave.