DoD Stabilization of linear hyperbolic PDEs on general cut-cell meshes

Abstract

Standard numerical methods for hyperbolic PDEs require for stability a CFL-condition which implies that the time step size depends on the size of the elements of the mesh. On cut-cell meshes, elements can become arbitrarily small and thus the time step size cannot take the size of small cut-cells into account but has to be chosen based on the background mesh elements.A remedy for this is the so called DoD (domain of dependence) stabilization for which several favorable theoretical and numerical properties have been shown in one and two space dimensions [4,9]. Up to now the method is restricted to stabilization of cut-cells with exactly one inflow and one outflow face, i.e. triangular cut-cells with a no-flow face (see [4]).We extend the DoD stabilization to cut-cells with multiple in-and outflow faces by properly considering the flow distribution inside the cut-cell. We further prove L 2 -stability for the semi-discrete formulation in space and present numerical results to validate the proposed extension. Copyright line will be provided by the publisher 1 IntroductionTo avoid the mesh generation process of complex geometries, cut-cell methods are an attractive alternative. The general idea is to start with a simple, e.g. structured, background mesh and to cut out the desired geometry. This results in a mesh with unstructured polyhedral cells, called cut-cells. Cut-cells can have an arbitrary shape and can become arbitrarily small, causing the small cell problem. To use explicit time stepping schemes for solving hyperbolic conservation laws, the time step size would need to be chosen based on the smallest cut-cell in the grid to ensure stability, which is in general not feasible.Developing solution approaches to the small cell problem in the context of discontinuous Galerkin (DG) schemes is a very recent research branch, including for example the work in [7,8,10]. In this contribution we focus on the domain of dependence (DoD) stabilization, which was introduced in [4] for the linear transport equation in one and two space dimensions and was extended to non-linear systems in one space dimension in [9]. It is based on a DG scheme in space to allow for higherorder approximations and possesses several desirable theoretical properties. Numerical results show the expected higher-order behavior in smooth flow and robustness around shocks.Up to now, the DoD stabilization in two dimensions has only been used to stabilize small triangular cut-cells for linear advection parallel to a ramp [4,11]. In this setup, the small stabilized cut-cells have exactly one inflow and one outflow face, which was exploited in the design of the stabilization. When moving to non-linear or coupled linear problems, this does not hold true anymore and one has to deal with multiple inflow and outflow faces.In this work, we take the first step in that direction by considering the linear advection equation on a cut-cell mesh with arbitrary flow directions, resulting in triangular cut-cells having 2 inflow and 1 outflow neighbor or reverse. As this ...