A Weakly Asymptotic Preserving Low Mach Number Scheme for the Euler Equations of Gas Dynamics
For computational fluid dynamics researchers, the paper highlights challenges in achieving asymptotic preserving properties for low Mach number flows, but the proposed scheme fails to maintain uniform consistency, making it an incremental step.
The paper proposes a low Mach number scheme for the Euler equations combining flux decomposition and explicit/implicit time discretization, but numerical experiments show the scheme requires stabilization for low Mach numbers, which breaks uniform asymptotic consistency and demands fine grids in the small Mach limit.
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.