DoD stabilization for higher-order advection in two dimensions

TL;DR

This paper introduces a domain of dependence (DoD) stabilization method for higher-order advection equations in 2D, addressing stability issues on cut cell meshes in a discontinuous Galerkin framework.

Contribution

It extends DoD stabilization to 2D higher-order advection problems, providing stability analysis and convergence results for arbitrary polynomial degrees.

Findings

Achieves $L^2$ stability for the stabilized scheme.

Demonstrates convergence orders of $p+1$ in $L^1$ norm.

Shows convergence orders between $p+\frac{1}{2}$ and $p+1$ in $L^{\infty}$ norm.

Abstract

When solving time-dependent hyperbolic conservation laws on cut cell meshes one has to overcome the small cell problem: standard explicit time stepping is not stable on small cut cells if the time step is chosen with respect to larger background cells. The domain of dependence (DoD) stabilization is designed to solve this problem in a discontinuous Galerkin framework. It adds a penalty term to the space discretization that restores proper domains of dependency. In this contribution we introduce the DoD stabilization for solving the advection equation in 2d with higher order. We show an $L^2$ stability result for the stabilized semi-discrete scheme for arbitrary polynomial degrees $p$ and provide numerical results for convergence tests indicating orders of $p+1$ in the $L^1$ norm and between $p+\frac 1 2$ and $p+1$ in the $L^{\infty}$ norm.