All speed scheme for the low mach number limit of the Isentropic Euler equation

TL;DR

This paper introduces an all-speed numerical scheme for the Isentropic Euler equations that remains accurate and stable across all Mach numbers, effectively capturing the incompressible limit without restrictive mesh constraints.

Contribution

The paper proposes a novel semi-implicit time discretization method that handles low Mach number regimes efficiently within a first-order Lax-Friedrich scheme.

Findings

The scheme accurately captures the incompressible limit as Mach number tends to zero.

It suppresses nonphysical oscillations compared to previous methods.

Numerical tests demonstrate the scheme's effectiveness across different Mach regimes.

Abstract

An all speed scheme for the Isentropic Euler equation is presented in this paper. When the Mach number tends to zero, the compressible Euler equation converges to its incompressible counterpart, in which the density becomes a constant. Increasing approximation errors and severe stability constraints are the main difficulty in the low Mach regime. The key idea of our all speed scheme is the special semi-implicit time discretization, in which the low Mach number stiff term is divided into two parts, one being treated explicitly and the other one implicitly. Moreover, the flux of the density equation is also treated implicitly and an elliptic type equation is derived to obtain the density. In this way, the correct limit can be captured without requesting the mesh size and time step to be smaller than the Mach number. Compared with previous semi-implicit methods, nonphysical oscillations can be suppressed. We develop this semi-implicit time discretization in the framework of a first order local Lax-Friedrich (LLF) scheme and numerical tests are displayed to demonstrate its performances.