Progress in all-Mach number flows and well-balanced methods for the compressible Euler equations

Abstract

Many astrophysical systems feature flows which are modeled by the multi-dimensional Euler equations. We discuss two aspects. (1) For the homogeneous Euler equations we look at flow in the low Mach number regime. Here for a conventional finite volume discretization one has excessive dissipation in this regime. We identify inconsistent scaling with low Mach number of numerical flux function as the origin of this problem. We propose a new flux preconditioner that ensures the correct scaling. We demonstrate that our new method is capable of representing flows down to Mach numbers of 10e-10. This is joint work with Wasilij Barsukow, Philipp Edelmann and Fritz Röpke. (2) For the Euler equations with gravity we seek well-balanced methods. We describe a numerical discretization of the compressible Euler equations with a gravitational potential. A pertinent feature of the solutions to these inhomogeneous equations is the special case of stationary solutions with zero velocity, described by a nonlinear PDE, whose solutions are called hydrostatic equilibria. We present a well-balanced method, for which we can ensure robustness, accuracy and stability, since it satisfies discrete entropy inequalities. This is joint work with Christophe Berthon, Praveen Chandrashekar, Vivien Desveaux and Markus Zenk