HyperCut – Stabilized DG schemes for hyperbolic conservation laws on cut cell meshes

Basic data for this project

Description

The goal of this project is to develop new tools for solving time-dependent, first order hyperbolic conservation laws, in particular the compressible Euler equations, on complex shaped domains.In practical applications, mesh generation is a major issue. When dealing with complicated geometries, the construction of corresponding body-fitted meshes is a very involved and time-consuming process.In this proposal, we will consider a different approach: In the last two decades so called cut cell methods have gained a lot of interest, as they reduce the burden of the meshing process. The idea is to simply cut the geometry out of a Cartesian background mesh. Theresulting cut cells can have various shapes and are not bounded from below in size. Compared to body-fitted meshes, this approach is fully automatic and much cheaper. However, standard explicit schemes are typically not stable when the time step is chosen with respect to the background mesh and does not reflect the size of small cut cells. Thisis referred to as the small cell problem.In the setting of standard meshes, both Finite Volume (FV) and Discontinuous Galerkin (DG) methods have been used successfully for solving non-linear hyperbolic conservation laws. For FV schemes, there already exist several approaches for extending thesemethods to cut cell meshes and overcoming the small cell problem while keeping the explicit time stepping. For DG schemes, this is not the case.The goal of this proposal is to develop stable DG schemes for solving time-dependent hyperbolic conservation laws, in particular the compressible Euler equations, on cut cell meshes using explicit time stepping.We particularly aim at a method that(1) solves the small cell problem and permits explicit time stepping,(2) preserves mass conservation,(3) is high-order along the cut cell boundary, where many important quantities are evaluated,(4) satisfies theoretical properties such as monotonicity and TVDM stability for model problems,(5) works for non-linear hyperbolic conservation laws, in particular the compressible Euler equations,(6) is robust in the presence of shocks or discontinuities,(7) and sufficiently simple to be implemented in higher dimensions.We base the spatial discretization on a DG approach to enable high accuracy. We plan to develop new stabilization terms to overcome the small cell problem for this setup. The starting point for this proposal is our recent publication for stabilizing a DG discretizationfor linear advection using piecewise linear polynomials. We will extend these results in different directions, namely to non-linear problems, including the compressible Euler equations, and to higher order, in particular to piecewise quadratic polynomials.We will implement these methods using the software framework DUNE and publish our code as open-source.