Speaker: Dr. Sigrun Ortleb, University of Kassel, homepage (guest of Willem Hundsdorfer)
Title: On Patankar-type time integration preserving non-negative water height within a discontinuous Galerkin shallow water code
Abstract: Discontinuous Galerkin(DG) methods are a modern and popular class of numerical methods especially for computationally intensive fluid dynamics calculations. Their popularity is due to the fact that DG methods allow for high order approximations in combination with high flexibility – e.g. in choosing different polynomial degrees on neighbouring elements. In this talk, we consider the application of the DG scheme on unstructured triangular grids to hyperbolic conservation laws. Briefly, we will show their connection to summation-by-parts(SBP) operators which posess very convenient stability properties.
We then focus on the application of the DG scheme to shallow water flows with non-flat bottom topography. In particular, the DG scheme then has to guarantee non-negativity of the water height. For locally refined grids at wet/dry interfaces, the stability and positivity requirements of explicit time integration unfortunately lead to rather restrictive time step constraints. However, the non-negativity requirement usually restricts the time step in the implicit case as well. In this context, we consider modified Patankar-type time integration methods which preserve non-negativity of the water height for any time step size.
