High order weight-adjusted discontinuous Galerkin methods for wave propagation on moving curved meshes

TL;DR

This paper develops high order weight-adjusted discontinuous Galerkin methods for wave equations on moving curved meshes, ensuring energy stability and high accuracy without costly mass matrix inversions, verified through numerical experiments.

Contribution

It introduces a novel weight-adjusted approximation within DG methods for wave problems on moving curved meshes, enhancing efficiency and stability.

Findings

Achieves high order accuracy verified by numerical experiments.

Ensures energy stability up to a small converging term.

Avoids mass matrix inversion at each time step.

Abstract

This paper presents high order accurate discontinuous Galerkin (DG) methods for wave problems on moving curved meshes with general choices of basis and quadrature. The proposed method adopts an arbitrary Lagrangian-Eulerian (ALE) formulation to map the acoustic wave equation from the time-dependent moving physical domain onto a fixed reference domain. For moving curved meshes, weighted mass matrices must be assembled and inverted at each time step when using explicit time stepping methods. We avoid this step by utilizing an easily invertible weight-adjusted approximation. The resulting semi-discrete weight-adjusted DG scheme is provably energy stable up to a term which converges to zero with the same rate as the optimal $L^2$ error estimate. Numerical experiments using both polynomial and B-spline bases verify the high order accuracy and energy stability of proposed methods.