Study of a New Asymptotic Preserving Scheme for the Euler System in the Low Mach Number Limit

This paper deals with the discretization of the compressible Euler system for all Mach number regimes. For highly subsonic flows, since acoustic waves are very fast compared to the velocity of the fluid, the gas can be considered as incompressible. From the numerical point of view, when the Mach number tends to zero, the classical Godunov type schemes present two main drawbacks: they lose consistency, and they suffer severe numerical constraints for stability to be guaranteed since the time step must follow the acoustic wave speed. In this work, we propose and analyze a new uniformly stable and consistent scheme for all Mach number flows, from compressible to incompressible regimes, stability being only related to the flow speed. We propose two space discretizations valid for all regimes. We present several one and two dimensional simulations which show the AP behavior of our schemes, and we observe their $L^\infty$ and/or $L^2$ stability in all regimes