Numerical Approximation of the Navier-Stokes Equations with Several Independent Specific Entropies

Abstract

The present work is devoted to the numerical approximation of the solutions of convective-diffusive systems which can be understood as natural extensions of the classical Navier-Stokes (NS) equations. Solutions of these systems are governed by N (N ≧ 2) independent pressure laws or equivalently N independent specific entropies. Several models from the physics actually enter the present framework. These extended forms are seen to share many properties with the usual NS equations (i.e. when N = 1) but with the very difference that, generally speaking, they cannot be recast in full conservation form. This property is known to make the definition of the endpoints of the Travelling Wave (TW) solutions sensitive with respect to the shape of the diffusive tensor (see [7]). Motivated by [1] devoted to N = 2, here we exhibit (N - 1) Generalized Rankine-Hugoniot (GRH) conditions that reflect this sensitivity. We underline the reason why classical algorithms can only fail in satisfying these GRE relations and we show how to enforce them for validity at the discrete level when extending [1] to N ≧ 2 general pressure laws. Several numerical evidences are proposed.