Weight-adjusted discontinuous Galerkin methods: wave propagation 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 $L^2$ inner product. In applications where the wavespeed varies spatially at a subelement scale, these matrices are distinct over each element, necessitating additional storage. In this work, we propose a weight-adjusted DG (WADG) method that reduces storage costs by replacing the weighted $L^2$ 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.