Time-Implicit Approximation of the Multipressure Gas Dynamics Equations in Several Space Dimensions

Abstract

The present work is devoted to the numerical approximation of the solutions of the inviscid limit of multi-pressure Navier-Stokes (NS) equations in several space dimensions. The nonconservative form of the Euler-like limit model makes the shock solutions to be sensitive with respect to the underlying small scales, and then challenging their numerical approximation. In particular, classical algorithms fails in producing good numerical results. Here we are mainly concerned with (large time stepping) implicit numerical strategies. We first exhibit a set of generalized jump conditions satisfied by the shock solutions and well-suited to derive a time-implicit scheme. We then devise a linearized time-implicit solver for the sake of efficiency. This solver is shown to preserve the positivity of each internal energy i provided that the total internal energy stays positive.

A Low Cost Semi-implicit Low-Mach Relaxation Scheme for the Full Euler Equations

“…Implicit schemes have been designed from the Suliciu relaxation method, [3,2]. The Suliciu relaxation system [12] writes…”

“…This whole scheme is detailed in appendix A. Note that another strategy has been proposed in [3,2] for the implicit treatment of the acoustic system. It is based on the diagonalization of the (Π, u) subsystem and the resolution of the two resulting transport equations with an implicit solver.…”

“…In particular this means to well resolve the flow scale (close to incompressible), but also to describe as much as possible the acoustic waves. Under the assumption (3) and rescaling the equations so as to have velocities of order unity, we classically have that ρ, u, T remain bounded. For times t at the flow scale a leading expansion is then…”

Implicit discretization of Lagrangian gas dynamics

“…where C v is the thermal capacity at constant volume, γ > 1 is the adiabatic index, e = E − 1 2 u 2 is the internal energy density, T is the temperature and c is the speed of sound given by c 2 = ∂p ∂ρ . The justification of using a prediction step based on the discretization of (2) comes from ideas in the work of Chalons, Coquel and Marmignon [5]. It will be explained in details in this article.…”

“…The predictor-corrector scheme is an adaptation of ideas from the article [5] which is dedicated to solve the Euler equations (6) in Eulerian coordinates…”