Monotonicity considerations for stabilized DG cut cell schemes for the unsteady advection equation

TL;DR

This paper investigates monotonicity conditions for stabilized discontinuous Galerkin schemes on cut cell meshes, addressing stability issues caused by small cell problems in unsteady advection equations.

Contribution

It extends previous stabilization methods by analyzing monotonicity conditions, enhancing the understanding of stability for DG schemes on complex meshes.

Findings

Stabilized schemes are stable in 1D and 2D.

Monotonicity can be achieved with specific stabilization parameters.

The approach improves stability without sacrificing accuracy.

Abstract

For solving unsteady hyperbolic conservation laws on cut cell meshes, the so called small cell problem is a big issue: one would like to use a time step that is chosen with respect to the background mesh and use the same time step on the potentially arbitrarily small cut cells as well. For explicit time stepping schemes this leads to instabilities. In a recent preprint [arXiv:1906.05642], we propose penalty terms for stabilizing a DG space discretization to overcome this issue for the unsteady linear advection equation. The usage of the proposed stabilization terms results in stable schemes of first and second order in one and two space dimensions. In one dimension, for piecewise constant data in space and explicit Euler in time, the stabilized scheme can even be shown to be monotone. In this contribution, we will examine the conditions for monotonicity in more detail.