Mathematikgebäude, Raum 1011

We propose and analyse space-time discontinuous Galerkin (DG) finite element methods for hyperbolic systems of conservation laws. To ensure entropy stability, the discretisation is performed in entropy variables and entropy-stable numerical fluxes are used. As solutions of hyperbolic conservation laws can develop discontinuities (shocks) in finite time, we include a streamline diffusion and a shock-capturing term in the formulation to suppress spurious oscillations in the vicinity of shocks. The approximate solutions are shown to converge to an entropy measure-valued solution for systems of conservation laws. The implicitness of the method does not give rise to a CFL condition. The method can therefore be applied to problems with multiple time scales such as low Mach number flows near the incompressible limit. Furthermore, the fully discrete space-time formulation allows a local refinement in space and time, which permits for instance to capture shocks more sharply. We employ residual-based as well as duality-based approaches to guide the refinement procedure. We have considered a large number of problems for instance for Burgers' equation, the wave equation, and the Euler equations in one or two spatial dimensions. These experiments demonstrate the robustness and the high-resolution properties of the schemes.

Dr. Andreas Hiltebrand

formerly Seminar for Applied Mathematics, ETH Zürich