Numerical methods for all-speed flows for the Euler equations including well-balancing of source terms

This thesis regards the numerical simulation of inviscid compressible ideal gases which are described by the Euler equations. We propose a novel implicit explicit (IMEX) relaxation scheme to simulate flows from compressible as well as near incompressible regimes based on a Suliciu-type relaxation model. The Mach number plays an important role in the design of the scheme, as it has great influence on the flow behaviour and physical properties of solutions of the Euler equations. Our focus is on an accurate resolution of the Mach number independent material wave. A special feature of our scheme is that it can account for the influence of a gravitational field on the fluid flow and is applicable also in small Froude number regimes. The time step of the IMEX scheme is constrained only by the eigenvalues of the explicitly treated part and is independent of the Mach number allowing for large time steps independent of the flow regime. In addition, the scheme is provably asymptotic preserving and well-balanced for arbitrary a priori known hydrostatic equilibria independently of the considered Mach and Froude regime. Also, the scheme preserves the positivity of density and internal energy throughout the simulation, it is well suited for physical applications. To increase the accuracy, a natural extension to second order is provided. The theoretical properties of the given schemes are numerically validated by various test cases performed on Cartesian grids in multiple space dimensions.