Numerical investigations of the compressible Navier-Stokes system

Abstract

In this paper we write, analyze and experimentally compare three different numerical schemes dedicated to the one dimensional barotropic Navier-Stokes equations: a staggered scheme based on the Rusanov one for the inviscid (Euler) system,a staggered pseudo-Lagrangian scheme in which the mesh “follows” the fluid,the Eulerian projection (on a fixed mesh) of the preceding scheme. All these schemes only involve the resolution of linear systems (all the nonlinear terms are solved in an explicit way). We propose numerical illustrations of their behaviors on particular solutions in which the density has discontinuities (hereafter called Hoff solutions). We show that the three schemes seem to converge to the same solutions, and we compare the evolution of the amplitude of the discontinuity of the numerical solution (with the pseudo-Lagrangian scheme) with the one predicted by Hoff and observe a good agreement.

A well-defined Godunov-type scheme in a Navier-Stokes model with a friction term

“…The Godunov scheme and its linearized extensions, known as Godunov-type schemes, are commonly used to illustrate finite volume schemes. Motivated by the articles [11], [1,2,4,6] and D. Hoff [19], we use the Godunovtype scheme to approximate the weak solutions of our problem (1.1). In [19], the existence of weak solutions has been established.…”