All Mach Number Second Order Semi-Implicit Scheme for the Euler Equations of Gasdynamics

TL;DR

This paper introduces a second-order semi-implicit scheme for the Euler equations of gas dynamics that is asymptotic preserving across all Mach numbers, reducing numerical diffusion and computational cost while maintaining accuracy.

Contribution

The paper develops a novel all Mach number finite volume scheme with semi-implicit discretization that simplifies flux computation and avoids iterative solvers, ensuring stability and accuracy.

Findings

Effective in both compressible and incompressible regimes

Second order accuracy in space and time achieved

CFL condition independent of Mach number

Abstract

This paper presents an asymptotic preserving (AP) all Mach number finite volume shock capturing method for the numerical solution of compressible Euler equations of gas dynamics. Both isentropic and full Euler equations are considered. The equations are discretized on a staggered grid. This simplifies flux computation and guarantees a natural central discretization in the low Mach limit, thus dramatically reducing the excessive numerical diffusion of upwind discretizations. Furthermore, second order accuracy in space is automatically guaranteed. For the time discretization we adopt an Semi-IMplicit/EXplicit (S-IMEX) discretization getting an elliptic equation for the pressure in the isentropic case and for the energy in the full Euler equations. Such equations can be solved linearly so that we do not need any iterative solver thus reducing computational cost. Second order in time is obtained by a suitable S-IMEX strategy taken from Boscarino et al. in [6]. Moreover, the CFL stability condition is independent of the Mach number and depends essentially on the fluid velocity. Numerical tests are displayed in one and two dimensions to demonstrate performances of our scheme in both compressible and incompressible regimes.