Positive and Entropy Stable Godunov-type Schemes for Gas Dynamics and MHD Equations in Lagrangian or Eulerian Coordinates

Abstract

Using the concept of simple Riemann solvers, we present entropic and positive Godunov-type schemes preserving contact discontinuities for both Lagrangian and Eulerian systems of gas dynamics and magnetohydrodynamics (MHD). On the one hand, for the Lagrangian form, we develop positive and entropic Riemann solvers which can be considered as a natural extension of Roe's solvers in which the sound speed is relaxed. On the other hand, for the Eulerian form, we are able to construct by two ways Godunov- type schemes based on Lagrangian simple Riemann solvers. The first method establishes a relation between the jump of the intermediate states and the second one between the intermediate states themselves.