Exact solution and the multidimensional Godunov scheme for the acoustic equations

The acoustic equations derived as a linearization of the Euler equations are a valuable system for studies of multi-dimensional solutions. Additionally they possess a low Mach number limit analogous to that of the Euler equations. Aiming at understanding the behaviour of the multi-dimensional Godunov scheme in this limit, first the exact solution of the corresponding Cauchy problem in three spatial dimensions is derived. The appearance of logarithmic singularities in the exact solution of the 4-quadrant Riemann Problem in two dimensions is discussed. The solution formulae are then used to obtain the multidimensional Godunov finite volume scheme in two dimensions. It is shown to be superior to the dimensionally split upwind/Roe scheme concerning its domain of stability and ability to resolve multi-dimensional Riemann problems. It is shown experimentally and theoretically that despite taking into account multi-dimensional information it is, however, not able to resolve the low Mach number limit.

Second-Order Invariant Domain Preserving ALE Approximation of Euler Equations

An invariant domain preserving arbitrary Lagrangian-Eulerian method for solving non-linear hyperbolic systems is developed. The numerical scheme is explicit in time and the approximation in space is done with continuous finite elements. The method is made invariant domain preserving for the Euler equations using convex limiting and is tested on various benchmarks.