A low Mach two-speed relaxation scheme for the compressible Euler equations with gravity

TL;DR

This paper introduces a novel low Mach number relaxation scheme for the Euler equations with gravity, capable of accurately capturing stationary states and the incompressible limit while ensuring positivity and entropy conditions.

Contribution

It develops a two-speed relaxation model with an approximate Riemann solver that preserves key physical and numerical properties, extending to second order accuracy.

Findings

Scheme preserves stationary solutions and low Mach limit.

Ensures positivity of density and internal energy.

Numerical experiments confirm theoretical properties.

Abstract

We present a numerical approximation of the solutions of the Euler equations with a gravitational source term. On the basis of a Suliciu type relaxation model with two relaxation speeds, we construct an approximate Riemann solver, which is used in a first order Godunov-type finite volume scheme. This scheme can preserve both stationary solutions and the low Mach limit to the corresponding incompressible equations. In addition, we prove that our scheme preserves the positivity of density and internal energy, that it is entropy satisfying and also guarantees not to give rise to numerical checkerboard modes in the incompressible limit. Later we give an extension to second order that preserves positivity, asymptotic-preserving and well-balancing properties. Finally, the theoretical properties are investigated in numerical experiments.