# Arbitrary Lagrangian-Eulerian discontinuous Galerkin method for conservation laws on moving simplex meshes

## TL;DR

This paper extends the ALE-DG method for conservation laws to multiple dimensions on simplex meshes, ensuring geometric conservation, stability, and high-order accuracy, validated through computational experiments.

## Contribution

The paper develops a multi-dimensional ALE-DG method on simplex meshes with proven stability, error estimates, and maximum principle, incorporating a limiter that preserves high-order accuracy.

## Key findings

- The method satisfies the geometric conservation law.
- L2-stability of the semi-discrete method is proven.
- Numerical experiments demonstrate stability and accuracy.

## Abstract

In Klingenberg, Schn\"ucke and Xia (Math. Comp. 86 (2017), 1203-1232) an arbitrary Lagrangian-Eulerian discontinuous Galerkin (ALE-DG) method to solve conservation laws has been developed and analyzed. In this paper, the ALE-DG method will be extended to several dimensions. The method will be designed for simplex meshes. This will ensure that the method satisfies the geometric conservation law, if the accuracy of the time integrator is not less than the value of the spatial dimension. For the semi-discrete method the L2-stability will be proven. Furthermore, an error estimate which provides the suboptimal (k+1/2) convergence with respect to the L-infinity-norm will be presented, when an arbitrary monotone flux is used and for each cell the approximating functions are given by polynomials of degree $k$. The two dimensional fully-discrete explicit method will be combined with the bound preserving limiter developed by Zhang, Xia and Shu in (J. Sci. Comput. 50 (2012), 29-62). This limiter does not affect the high order accuracy of a numerical method. Then, for the ALE-DG method revised by the limiter the validity of a discrete maximum principle will be proven. The numerical stability, robustness and accuracy of the method will be shown by a variety of two dimensional computational experiments on moving triangular meshes.