Weight-adjusted discontinuous Galerkin methods: matrix-valued weights and elastic wave propagation in heterogeneous media
This work provides a practical, high-order accurate numerical method for elastic wave propagation in heterogeneous media, which is important for geophysics and engineering applications.
The paper extends weight-adjusted discontinuous Galerkin (WADG) methods to handle matrix-valued weights for elastic wave propagation in heterogeneous media, achieving low storage, energy stability, and high-order accuracy. Numerical results confirm stability and high-order accuracy for several elastic wave problems.
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.