Mathematikgebäude, Seminarraum 1011

We discuss finite volume methods for hyperbolic pdes on Cartesian grids with embedded boundaries. Embedded boundary methods are very attractive for several reasons: The grid generation is simple even in the presence of complicated geometries. Furthermore, such an approach allows the use of regular Cartesian grid methods away from the embedded boundary, which are much simpler to construct and more accurate than unstructured grid methods. In embedded boundary grids with cut cells adjacent to the boundary, the cut cell volumes can be orders of magnitude smaller than a regular Cartesian grid cell volume. The use of standard difference procedures would lead to an unacceptable small integration timestep. Both accuracy and stability are issues that need to be addressed at these highly irregular cut cells adjacent to solid bodies. The goal is to construct a method which is stable for time steps that are appropriate for the regular part of the mesh and at the same time do not lead to a loss of accuracy. While previous finite volume cut cell methods have been constructed to obtain a second order accurate method, we are interested in higher order schemes. On the regular part of the grid we use the so called active flux method, a new finite volume method proposed by Roe et al. In my talk, after a short review of finite volumes methods, the active flux method will be explained in detail. 1d and 2d considerations will lead us to preliminiary results on the use of the active flux method for cut cells. Finally, a possible extension to nonlinear systems of equations will be introduced and its use in cut cell methods will be discussed.

David Kerkmann

Heinrich Heine Universität Düsseldorf