Lax-Friedrichs and the Viscosity-Capillarity Criterion

Abstract

It has been shown by Lax some time ago that for hyperbolic conversation laws solutions obtained as limits of the Lax-Friedrichs finite difference scheme will actually satisfy an "entropy" admissibility criterion. The goal of this paper is to attempt to extend Lax's idea to a form which is amenable to mixed problems as well, e.g. the dynamics of a van der Waals fluid. Specifically, we compare shocks obtained by the Lax-Friedrichs scheme with those permitted by the viscosity-capillarity criterion of [2, 3]. We show that for isothermal motion it is expected that shocks produced by Lax-Friedrichs will satisfy the viscosity-capillarity criterion.