Provably fully discrete energy-stable and asymptotic-preserving scheme for barotropic Euler equations

TL;DR

This paper introduces a novel finite volume scheme for the barotropic Euler equations that guarantees energy stability and asymptotic-preserving properties in the low Mach number regime, with rigorous proofs and numerical validation.

Contribution

The paper presents a new implicit-explicit finite volume scheme that ensures entropy stability, positivity, and asymptotic consistency for low Mach number flows, which was not previously achieved.

Findings

The scheme is proven to be entropy stable and positivity-preserving.

Numerical experiments confirm the scheme's structure-preserving properties.

The method is asymptotic-preserving and effective for benchmark problems.

Abstract

We develop structure-preserving finite volume schemes for the barotropic Euler equations in the low Mach number regime. Our primary focus lies in ensuring both the asymptotic-preserving (AP) property and the discrete entropy stability. We construct an implicit-explicit (IMEX) method with suitable acoustic/advection splitting including implicit numerical diffusion that is independent of the Mach number. We prove the positivity of density, the entropy stability, and the asymptotic consistency of the fully discrete numerical method rigorously. Numerical experiments for benchmark problems validate the structure-preserving properties of the proposed method.