Error analysis of an asymptotic-preserving, energy-stable finite volume method for barotropic Euler equations

TL;DR

This paper introduces an energy-stable, asymptotic-preserving finite volume method for the Euler equations, providing rigorous error estimates and convergence analysis in both compressible and incompressible regimes.

Contribution

It develops a novel finite volume scheme with proven error bounds that are uniform across different Mach number regimes, ensuring reliable simulations from compressible to incompressible flows.

Findings

Error estimates are uniform in Mach number and discretisation parameters.

Convergence to strong solutions of Euler equations is rigorously established.

Numerical experiments validate the theoretical error bounds.

Abstract

We design an energy-stable and asymptotic-preserving finite volume scheme for the compressible Euler system. Using the relative energy framework, we establish rigorous error estimates that yield convergence of the numerical solutions in two distinct regimes. For a fixed Mach number $\varepsilon>0$, we derive error estimates between the numerical solutions and a strong solution of the compressible Euler system that are uniform with respect to the discretisation parameters, ensuring convergence as the underlying mesh is refined. In the low Mach number regime, we analyse the error between the numerical solutions and a strong solution of the incompressible Euler system and obtain asymptotic error estimates that are uniform in $\varepsilon$ and the discretisation parameters. These results imply convergence of the numerical solutions toward a strong solution of the incompressible Euler system as $\varepsilon$, and the discretisation parameters simultaneously tend to zero. Numerical experiments are presented to validate the theoretical analysis.