# A stabilized DG cut cell method for discretizing the linear transport equation

## TL;DR

This paper introduces a new stabilization technique for the discontinuous Galerkin method applied to the linear transport equation on cut cell meshes, enabling explicit time stepping without stability issues.

## Contribution

The authors develop and analyze stabilization terms that allow explicit time stepping on cut cell meshes in DG discretizations of the linear transport equation.

## Key findings

- Monotonicity proven for piecewise constant polynomials in 1D
- Total variation diminishing stability shown for piecewise linear polynomials
- Numerical results confirm theoretical stability and accuracy

## Abstract

We present new stabilization terms for solving the linear transport equation on a cut cell mesh using the discontinuous Galerkin (DG) method in two dimensions with piecewise linear polynomials. The goal is to allow for explicit time stepping schemes, despite the presence of cut cells. Using a method of lines approach, we start with a standard upwind DG discretization for the background mesh and add penalty terms that stabilize the solution on small cut cells in a conservative way. Then, one can use explicit time stepping, even on cut cells, with a time step length that is appropriate for the background mesh. In one dimension, we show monotonicity of the proposed scheme for piecewise constant polynomials and total variation diminishing in the means stability for piecewise linear polynomials. We also present numerical results in one and two dimensions that support our theoretical findings.