An Asymptotic Preserving and Energy Stable Scheme for the Barotropic Euler System in the Incompressible Limit

An asymptotic preserving and energy stable scheme for the barotropic Euler system under the low Mach number scaling is designed and analysed. A velocity shift proportional to the pressure gradient is introduced in the convective fluxes, which leads to the dissipation of mechanical energy and the entropy stability at all Mach numbers. The resolution of the semi-implicit in time and upwind in space fully-discrete scheme involves two steps: solution of an elliptic problem for the density and an explicit evaluation for the velocity. The proposed scheme possess several physically relevant attributes, such as the positivity of density, the entropy stability and the consistency with the weak formulation of the continuous Euler system. The AP property of the scheme, i.e.\ the boundedness of the mesh parameters with respect to the Mach number and its consistency with the incompressible limit system, is shown rigorously. The results of extensive case studies are presented to substantiate the robustness and efficacy of the proposed scheme as well as the theoretical claims.