Communications in Mathematical Sciences

Volume 21 (2023)

Number 8

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

Pages: 2213 – 2246



Claudius Birke (Fakultät für Mathematik und Informatik, Universität Würzburg, Germany)

Christophe Chalons (Laboratoire de Mathématiques de Versailles, Université Versailles Saint-Quentin-en-Yvelines (UVSQ), CNRS, Université Paris-Saclay, Versailles, France)

Christian Klingenberg (Fakultät für Mathematik und Informatik, Universität Würzburg, Germany)


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.


Euler equations, finite volume methods, relaxation, well-balancing, low Mach, asymptotic-preserving, entropy satisfying, checkerboard modes, positivity preserving

2010 Mathematics Subject Classification

65M08, 76M12

Claudius Birke acknowledges the support by the German Research Foundation (DFG) under the project no. KL 566/22-1. All three authors acknowledge the project “Bayerisch-Französische Hochschulzentrum FK40 2019” which supported this work.

Received 31 October 2022

Received revised 23 March 2023

Accepted 24 March 2023

Published 15 November 2023