High-order accurate numerical methods for hyperbolic partial differential equations on Cartesian grids with embedded geometry

• Applicant: Professorin Dr. Christiane Helzel
• Subject Area: Mathematics
• Term: since 2016
• Project identifier: Deutsche Forschungsgemeinschaft (DFG) - Project number 325695158

This project is devoted to the construction of stable, conservative and third order accurate Cartesian grid cut cell finite volume methods for hyperbolic partial differential equations.Many applications require the solution of hyperbolic problems in complicated geometries. The generation of a computational mesh is a basic task in every finite volume method. Traditionally, either a structured body-fitted grid or a completely unstructured grid is used. Both of these approaches require a significant effort in mesh generation. An alternative approach is the use of a Cartesian grid cut cell method. This approach cuts solid bodies out of a background Cartesian mesh. Such an approach simplifies the grid generation. Away from the boundary, a cut cell approach allows the use of high-resolution Cartesian grid finite volume methods. These methods are typically easier than unstructured grid methods of the same accuracy. However, the cut cells need a special treatment in order to maintain stability of explicit methods in computations where some of the cut cells might be orders of magnitude smaller than regular grid cells. Beside the stability issue it turns out that maintaining accuracy along the embedded boundary is much more challenging. In this project we develop the first third order accurate Cartesian grid cut cell method. Away from the boundary we use the Active Flux method, a new third order accurate finite volume method with a compact stencil that is based on the use of cell average values and point values of the conserved quantity. At cut cells this method needs to be modified in order to maintain stability as well as accuracy.

• DFG Programme: Research Grants