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

TL;DR

This paper extends the DoD stabilization method for hyperbolic PDEs on cut-cell meshes to handle complex cells with multiple flow faces, ensuring stability without restrictive time step constraints.

Contribution

The paper introduces a novel extension of DoD stabilization for complex cut-cells with multiple in- and out-flow faces, maintaining stability in hyperbolic PDE simulations.

Findings

Proves L2-stability for the extended method

Validates the approach with numerical experiments

Enables stable simulations on more complex cut-cell geometries

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. 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. We extend the DoD stabilization to cut-cells with multiple in- and out-flow faces by properly considering the flow distribution inside the cut-cell. We further prove L2-stability for the semi-discrete formulation in space and present numerical results to validate the proposed extension.