HyperCut II -- Stabilized higher order DG schemes for hyperbolic conservation laws on cut cell meshes

• Applicants: Professor Dr. Christian Engwer; Professor Dr. Stefan Turek, since 5/2022
• Subject Area: Mathematics
• Term: since 2020
• Project identifier: Deutsche Forschungsgemeinschaft (DFG) - Project number 439956613

In this project, we continue the work of the DFG funded project HyperCut in developing stabilized Discontinuous Galerkin (DG) schemes for hyperbolic conservation laws on cut cell meshes. Cut cell meshes are an alternative to, e.g., body-fitted meshes; the given geometry is simply cut out of a structured background mesh. Where the objects intersect with the underlying mesh, this results in so-called cut cells, which can become arbitrarily small. When solving time-dependent hyperbolic conservation laws using explicit time stepping, one faces the so-called small cell problem: standard schemes are not stable on the cut cells and their neighbors if the time step is chosen according to the larger background cells. Solving this issue in the context of DG schemes is a very recent research area and only very few solution approaches exist, including our Domain-of-Dependence (DoD) stabilization. The DoD stabilization is designed to restore the proper domains of dependence of outflow neighbors of cut cells and thereby allows to use explicit time stepping with a time step length that is independent of the size of small cut cells. The DoD stabilization consists of two penalty terms. One is responsible for redistributing the mass between the small cut cells and their neighbors appropriately. The other one redistributes the mass within the cells. In this project we will extend the DoD stabilization to non-linear systems in two and three dimensions, aiming for higher order accuracy. We will design the stabilization to ensure L2 stability for scalar problems and entropy stability for Euler equations. We will also address practical issues such as curved boundaries and limiting on cut cells. All the implementation will be done within the DUNE framework to enable state-of-the-art and stable simulations for the Euler equations in three dimensions on domains with complex geometries and will be publicly available.

• DFG Programme: Research Grants
• Co-Investigator: Professor Dr. Hendrik Ranocha
• Ehemalige Antragstellerin: Professorin Dr. Sandra May, Ph.D., until 4/2022