A numerical scheme for the compressible low-Mach number regime of ideal fluid dynamics

TL;DR

This paper analyzes a Roe-based numerical scheme designed for accurately simulating low Mach number flows in ideal fluid dynamics, demonstrating its stability and effectiveness down to extremely low Mach numbers (~1e-10).

Contribution

It provides a detailed analysis of the scheme's properties, stability, and performance, extending the applicability of Roe-based methods to very low Mach regimes.

Findings

Scheme accurately captures flows down to Mach ~1e-10

Linear stability analysis confirms scheme stability under explicit time integration

Implicit integration performance assessed via condition number evolution

Abstract

Based on the Roe solver a new technique that allows to correctly represent low Mach number flows with a discretization of the compressible Euler equations was proposed in Miczek et al.: New numerical solver for flows at various mach numbers. A&A 576, A50 (2015). We analyze properties of this scheme and demonstrate that its limit yields a discretization of the continuous limit system. Furthermore we perform a linear stability analysis for the case of explicit time integration and study the performance of the scheme under implicit time integration via the evolution of its condition number. A numerical implementation demonstrates the capabilities of the scheme on the example of the Gresho vortex which can be accurately followed down to Mach numbers of ~1e-10 .