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

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 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.

On discretely entropy stable weight-adjusted discontinuous Galerkin methods: curvilinear meshes

“…Applying this correction to the right hand side of (31) then yields a scheme which locally and globally conserves mean values of the conservative variables. This approach is applicable to an arbitrary weight, and can be generalized to matrix-valued weights as well [21]. Moreover, using Theorem 6 in [3], one can show that the L 2 norm of the difference u WADG −u WADG is O(h 2N +1 ), and does not affect high order accuracy.…”

Bernstein-Bézier weight-adjusted discontinuous Galerkin methods for wave propagation in heterogeneous media

“…where V , Σ are constructed by concatenating Σ i , V i into single vectors, respectively, and F v , F σ are vectors representing the velocity and stress numerical fluxes. We note that this formulation is energy stable and high order accurate for elastic wave propagation in either isotropic or aniostropic heterogeneous media [21].…”

“…From these plots, we observe that the convergence rate It should be noted that these rates of convergence are better than those suggested by an initial error analysis. It is straightforward to extend the error analysis of [1,21] to accomodate approximations of c 2 ∈ P M . However, this extension predicts that, when gence by one order for each degree past M = 1.…”

Multi-patch discontinuous Galerkin isogeometric analysis for wave propagation: Explicit time-stepping and efficient mass matrix inversion

“…The dissipation resulting from the stabilization terms provides a natural way to damp spurious non-conforming components of the solution in time-domain simulations [21]. Similar energy stable skew-symmetric formulations can be constructed for electromagnetics and elastodynamics [22,23]. Dirichlet boundary conditions on pressure are enforced in an energy-stable fashion by specifying the jump of p on boundary faces…”

“…It is possible to offset the sequential nature of the Cholesky backsolve by parallelizing over each of these one-dimensional mass inversions. However, this approach also requires significantly more storage, which can be problematic on many-core architectures such as Graphics Processing Units[23].…”