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

在这个项目中，我们继续DFG资助的项目HyperCut的工作，在切割细胞网格上开发双曲守恒律的稳定间断伽辽金（DG）格式。切割细胞网格是一种替代方案，例如，适合身体的网格；给定的几何体是简单地从一个结构化的背景网格中切割出来的。当物体与底层网格相交时，就会产生所谓的切割细胞，它可以变得任意小。当使用显式时间步进求解时间相关的双曲守恒律时，会面临所谓的小细胞问题：如果根据较大的背景细胞选择时间步长，则标准方案在切割细胞及其邻近细胞上不稳定。在DG方案的背景下解决这个问题是一个非常新的研究领域，只有很少的解决方法存在，包括我们的依赖域（DoD）稳定化。DoD稳定化设计用于恢复切割细胞的流出邻居的适当依赖域，从而允许使用与小切割细胞大小无关的显式时间步长。国防部稳定由两个惩罚项组成。其中一个负责在小切割细胞和邻近细胞之间适当地重新分配质量。另一个在细胞内重新分配物质。在这个项目中，我们将把DoD稳定化扩展到二维和三维的非线性系统，以达到更高的阶精度。我们将设计稳定性以确保标量问题的L2稳定性和欧拉方程的熵稳定性。我们还将讨论实际问题，如弯曲边界和对切割细胞的限制。所有的实现都将在DUNE框架内完成，以便在具有复杂几何形状的域上实现最先进和稳定的三维欧拉方程模拟，并将公开提供。

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.