Weight-adjusted discontinuous Galerkin methods: matrix-valued weights and elastic wave propagation in heterogeneous media

TL;DR

This paper extends weight-adjusted discontinuous Galerkin methods to matrix-valued weights, enabling stable, high-order accurate elastic wave simulations in complex heterogeneous media using efficient, low-storage algorithms.

Contribution

The work introduces matrix-valued weight adjustments into DG methods for elastic waves, enhancing stability and accuracy in heterogeneous media simulations.

Findings

WADG methods are stable and highly accurate for elastic wave problems.

Numerical results confirm the effectiveness of matrix-valued weights in complex media.

The approach reduces storage requirements while maintaining high-order convergence.

Abstract

Weight-adjusted inner products are easily invertible approximations to weighted $L^2$ inner products. These approximations can be paired with a discontinuous Galerkin (DG) discretization to produce a time-domain method for wave propagation which is low storage, energy stable, and high order accurate for arbitrary heterogeneous media and curvilinear meshes. In this work, we extend weight-adjusted DG (WADG) methods to the case of matrix-valued weights, with the linear elastic wave equation as an application. We present a DG formulation of the symmetric form of the linear elastic wave equation, with upwind-like dissipation incorporated through simple penalty fluxes. A semi-discrete convergence analysis is given, and numerical results confirm the stability and high order accuracy of WADG for several problems in elastic wave propagation.