Analysis of an Asymptotic Preserving Low Mach Number Accurate IMEX-RK Scheme for the Wave Equation System

TL;DR

This paper presents an analysis of an asymptotic preserving IMEX-RK finite volume scheme for the wave equation in the low Mach number limit, ensuring stability, accuracy, and convergence across Mach regimes.

Contribution

It introduces a novel IMEX-RK scheme that maintains accuracy and stability uniformly as Mach number approaches zero, with rigorous proofs and numerical validation.

Findings

Uniform second order convergence at low Mach numbers

Existence and uniqueness of the numerical solution

Scheme's stability and asymptotic preserving properties

Abstract

In this paper the analysis of an asymptotic preserving (AP) IMEX-RK finite volume scheme for the wave equation system in the zero Mach number limit is presented. The accuracy of a numerical scheme at low Mach numbers is its ability to maintain the solution close to the incompressible solution for all times, and this can be formulated in terms of the invariance of a space of constant densities and divergence-free velocities. An IMEX-RK methodology is employed to obtain a time semi-discrete scheme, and a space-time fully-discrete scheme is derived by using standard finite volume techniques. The existence of a unique numerical solution, its uniform stability with respect to the Mach number, the AP property, and the accuracy at low Mach numbers are established for both time semi-discrete, and space-time fully-discrete schemes. Extensive numerical case studies confirm uniform second order convergence of the scheme with respect to the Mach number, and all the above-mentioned properties.