An all speed second order well-balanced IMEX relaxation scheme for the Euler equations with gravity

TL;DR

This paper introduces a second-order, well-balanced IMEX relaxation scheme for Euler equations with gravity, capable of handling low Mach flows with scale-independent time steps and preserving key physical properties.

Contribution

The paper develops a novel second-order IMEX relaxation scheme that is well-balanced, positivity-preserving, and asymptotic preserving for Euler equations with gravity, including low Mach regimes.

Findings

Scheme is well-balanced and positivity-preserving.

Numerical tests validate scale-independent diffusion.

Method accurately captures low Mach limit behavior.

Abstract

We present an implicit-explicit well-balanced finite volume scheme for the Euler equations with a gravitational source term which is able to deal also with low Mach flows. To visualize the different scales we use the non-dimensionalized equations on which we apply a pressure splitting and a Suliciu relaxation. On the resulting model, we apply a splitting of the flux into a linear implicit and an non-linear explicit part that leads to a scale independent time-step. The explicit step consists of a Godunov type method based on an approximative Riemann solver where the source term is included in the flux formulation. We develop the method for a first order scheme and give an extension to second order. Both schemes are designed to be well-balanced, preserve the positivity of density and internal energy and have a scale independent diffusion. We give the low Mach limit equations for well-prepared data and show that the scheme is asymptotic preserving. These properties are numerically validated by various test cases.