A Structure-Preserving Scheme for the Euler System with Potential Temperature Transport

TL;DR

This paper introduces a novel all-speed finite volume scheme for the Euler equations with potential temperature transport, ensuring stability, positivity, and accuracy across all Mach number regimes, including the low Mach limit.

Contribution

The authors develop a semi-implicit, asymptotic preserving scheme that maintains physical positivity and consistency with both compressible and incompressible limits.

Findings

The scheme is stable and accurate across a wide Mach number range.

Numerical tests confirm robustness in capturing complex flow features.

The method preserves key physical and mathematical structures.

Abstract

We consider the compressible Euler equations with potential temperature transport, a system widely used in atmospheric modelling to describe adiabatic, inviscid flows. In the low Mach number regime, the equations become stiff and pose significant numerical challenges. We develop an all-speed, semi-implicit finite volume scheme that is asymptotic preserving (AP) in the low Mach limit and strictly positivity preserving for density and potential temperature. The scheme ensures stability and accuracy across a broad range of Mach numbers, from fully compressible to nearly incompressible regimes. We rigorously establish consistency with both the compressible system and its incompressible, density-dependent limit. Numerical experiments confirm that the method robustly captures complex flow features while preserving the essential physical and mathematical structures of the model.