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HyperCut项目的工作，在切割网格上开发双曲型守恒律的稳定化不连续Galerkin(DG)格式。切割单元网格是例如身体贴合网格的替代方案；给定的几何图形简单地从结构化背景网格中切割出来。在对象与底层网格相交的地方，这会导致所谓的剪切单元，它可以变得任意小。在用显式时间步长求解含时双曲型守恒律时，会遇到所谓的小网格问题：如果根据较大的背景单元来选择时间步长，则标准格式在截断单元及其相邻单元上是不稳定的。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稳定化被设计为恢复切割单元的流出邻域的适当依赖域，从而允许使用与小切割单元的大小无关的时间步长的显式时间步长。国防部的稳定由两个惩罚条款组成。一个负责在小的切割细胞和它们的邻居之间适当地重新分配质量。另一种是在细胞内重新分配质量。在这个项目中，我们将把DOD稳定化推广到二维和三维的非线性系统，以达到更高的阶次精度。我们将设计稳定化以确保标量问题的L2稳定性和欧拉方程的熵稳定性。我们还将解决实际问题，如曲线边界和对切割单元格的限制。所有的实现都将在沙丘框架内完成，以便能够在具有复杂几何形状的区域上对三维欧拉方程进行最先进和稳定的模拟，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.