Single-Step Arbitrary Lagrangian-Eulerian Discontinuous Galerkin Method for 1-D Euler Equations

TL;DR

This paper introduces an explicit single-step discontinuous Galerkin method on moving grids using ALE for 1D Euler equations, improving accuracy and reducing dissipation.

Contribution

It develops a novel ALE-DG scheme that preserves constant states, incorporates grid refinement, and extends to higher order methods for better accuracy.

Findings

Reduces numerical dissipation with smoothed grid movement

Preserves constant states under mesh motion

Demonstrates high accuracy through test cases

Abstract

We propose an explicit, single step discontinuous Galerkin (DG) method on moving grids using the arbitrary Lagrangian-Eulerian (ALE) approach for one dimensional Euler equations. The grid is moved with the local fluid velocity modified by some smoothing, which is found to considerably reduce the numerical dissipation introduced by Riemann solvers. The scheme preserves constant states for any mesh motion and we also study its positivity preservation property. Local grid refinement and coarsening are performed to maintain the mesh quality and avoid the appearance of very small or large cells. Second, higher order methods are developed and several test cases are provided to demonstrate the accuracy of the proposed scheme.