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

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 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 semidiscrete convergence analysis is given, and numerical results confirm the stability and high-order accuracy of weight-adjusted DG for several problems in elastic wave propagation.