A Weakly Asymptotic Preserving Low Mach Number Scheme for the Euler Equations of Gas Dynamics

TL;DR

This paper introduces a low Mach number scheme for gas dynamics that combines flux decomposition and explicit/implicit discretization, aiming for asymptotic consistency and efficiency in simulations across Mach numbers.

Contribution

It develops a novel scheme that integrates Klein's flux decomposition with an explicit/implicit time discretization, addressing challenges in asymptotic preservation at low Mach numbers.

Findings

Scheme is asymptotically consistent but needs stabilization at low Mach numbers.

Stability issues affect asymptotic accuracy, requiring fine grids.

CFL condition depends only on non-stiff speeds, independent of Mach number.

Abstract

We propose a low Mach number, Godunov-type finite volume scheme for the numerical solution of the compressible Euler equations of gas dynamics. The scheme combines Klein's non-stiff/stiff decomposition of the fluxes (J. Comput. Phys. 121:213-237, 1995) with an explicit/implicit time discretization (Cordier et al., J. Comput. Phys. 231:5685- 5704, 2012) for the split fluxes. This results in a scalar second order partial differential equation (PDE) for the pressure, which we solve by an iterative approximation. Due to our choice of a crucial reference pressure, the stiff subsystem is hyperbolic, and the second order PDE for the pressure is elliptic. The scheme is also uniformly asymptotically consistent. Numerical experiments show that the scheme needs to be stabilized for low Mach numbers. Unfortunately, this affects the asymptotic consistency, which becomes non-uniform in the Mach number, and requires an unduly fine grid in the small Mach number limit. On the other hand, the CFL number is only related to the non-stiff characteristic speeds, independently of the Mach number. Our analytical and numerical results stress the importance of further studies of asymptotic stability in the development of AP (asymptotic preserving) schemes.