Structurally stable Riemann solutions

Abstract

We study the structure of solutions of Riemann problems for systems of two conservation laws. Such a solution comprises a sequence of elementary waves, viz., rarefaction and shock waves of various types; shock waves are required to have viscous profiles. We construct a Riemann solution by solving a system of equations characterizing its component waves. A Riemann solution is "structurally stable" if the number and types of its component waves are preserved when the initial data and the flux function are perturbed. Under the assumption that rarefaction waves and shock states lie in the stricly hyperbolic region, we characterize Riemann solutions for which the definition equations have maximal rank and we prove that such solutions are structurally stable. Structurally stable Riemann solutions cannot contain overcompressive shock waves, but they can contain transitional shock waves, including doubly sonic transitional shock waves, including doubly sonic transitional shock waves that have not been observed before. © 1996 Academic Press, Inc.