An all Mach number relaxation upwind scheme

Abstract

The present paper concerns the derivation of finite volume methods to approximate the weak solutions of the Euler equations within all Mach number regimes. To address such an issue, we develop a Suliciu relaxation type scheme. By adopting a relevant scaling according to the Mach number, the obtained numerical scheme is proved to be accurate in the sense that the numerical viscosity does not increase as soon as the Mach number tends to zero. Moreover, the obtained scheme is proved to be asymptotic preserving since the correct incompressible asymptotic regime is recovered in the limit of the Mach number to zero. In addition, the robustness of the method is established since both density and internal energy remain positive during the simulations. Several numerical experiments in 1D and 2D are performed to illustrate the relevance of the proposed low Mach number numerical scheme.