Weight-adjusted discontinuous Galerkin methods: wave propagation in heterogeneous media

TL;DR

This paper introduces a weight-adjusted discontinuous Galerkin method that reduces storage costs and maintains energy stability for wave propagation in media with spatially varying wavespeed.

Contribution

It proposes a novel weight-adjusted inner product for DG methods, improving efficiency in heterogeneous media without increasing storage requirements.

Findings

The method is energy stable.

It achieves comparable accuracy to traditional methods.

Numerical examples demonstrate effectiveness in heterogeneous media.

Abstract

Time-domain discontinuous Galerkin (DG) methods for wave propagation require accounting for the inversion of dense elemental mass matrices, where each mass matrix is computed with respect to a parameter-weighted L2 inner product. In applications where the wavespeed varies spatially at a sub-element scale, these matrices are distinct over each element, necessitating additional storage. In this work, we propose a weight-adjusted DG (WADG) method which reduces storage costs by replacing the weighted L2 inner product with a weight-adjusted inner product. This equivalent inner product results in an energy stable method, but does not increase storage costs for locally varying weights. A-priori error estimates are derived, and numerical examples are given illustrating the application of this method to the acoustic wave equation with heterogeneous wavespeed.