Weight-Adjusted Discontinuous Galerkin Methods: Wave Propagation in Heterogeneous Media

Abstract

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 sub-element scale, these matrices are distinct over each element, necessitating additional storage. In this work, we propose a weight-adjusted DG (WADG) method which 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.1. Introduction. Accurate numerical simulations of wave propagation through complex media are becoming increasingly important in seismology, especially as modern computational resources make the use of high fidelity subsurface models feasible for seismic imaging and full waveform inversion. A host of different numerical methods are currently in use, the most popular of which are high order finite difference methods [1]. While finite difference methods tend to perform excellently for simple geometries and smoothly varying data, their accuracy is degraded for heterogeneous media with interfaces or sharp gradients [2].In order to address these issues, high order finite element methods for wave propagation have been considered as alternatives to finite difference methods. A drawback of using continuous finite elements for time-domain simulations using explicit timestepping is the inversion of a global mass matrix system at each timestep. Spectral Element Methods (SEM) [3] address this issue by diagonalizing this mass matrix system through the use of mass-lumping, which co-locates interpolation nodes for Lagrange basis functions and Gauss-Legendre-Lobatto quadrature points. Since SEM is limited to unstructured hexahedral meshes, which are less geometrically flexible than tetrahedral meshes, triangular and tetrahedral mass-lumped spectral element methods have been investigated as alternatives [4,5,6]. However, due to a mismatch in the number of natural quadrature nodes and the dimension of polynomial approximation spaces on simplices, these methods necessitate adding additional nodes in the interior of the element to construct sufficiently accurate nodal points suitable for mass-lumping. Additionally, mass-lumpable nodal points on tetrahedra have only been determined for polynomial bases of degree four or less [4].High order discontinuous Galerkin (DG) methods have been considered as an alternative to Spectral Element Methods for seismic wave propagation [7,8,9,10]. Instead of using mass-lumping to arrive at a diagonal mass matrix, DG methods naturally induce a block diagonal mass matrix through the use of arbitrary-order approximation spaces which are disconti...

A weight-adjusted discontinuous Galerkin method for the poroelastic wave equation: Penalty fluxes and micro-heterogeneities

“…In all experiments, we follow [41] and compute the application of weight-adjusted mass matrices using a quadrature which is exact for polynomials of degree (2N + 1). Time integration is performed using the low-storage 4 th order five-stage Runge-Kutta scheme of Carpenter and Kennedy [42], and the time step is chosen based on the global estimate…”

Assessment of Hybrid High-Order methods on curved meshes and comparison with discontinuous Galerkin methods

“…The local linear problems involved in the computation of the local lifting operators (11) for the DG method, as well as in the computation of the potential reconstruction (14) and in the static condensation for the HHO method, are exactly solved exactly by means of Cholesky factorizations.…”

“…Denote by v a scalar-valued function on T whose regularity will be detailed in what follows. Recalling the definitions (13) of the interpolator, (14) of the potential reconstruction, and (4) of the elliptic projector…”

“…In the former reference, orthonormal bases are obtained by means of a modified Gram-Schmidt procedure to ensure numerical-stability at high-polynomial degrees, while in the latter the same goal is attained by incorporating the spatial variation of the element Jacobian into the physical basis functions. On the other hand, recent works by Chan et al [13,14] rely on reference frame polynomial spaces introducing weight-adjusted L 2 -inner products in order to recover high-order accuracy. HDG has been employed on meshes with curved boundaries, mainly in the context of compressible flow problems [31,28]; eXtended HDG with level-set description of interfaces has been recently investigated by Gurkan et al [27].…”

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

“…

This paper presents an efficient discontinuous Galerkin method to simulate wave propagation in heterogeneous media with sub-cell variations. This method is based on a weight-adjusted discontinuous Galerkin method (WADG), which achieves high order accuracy for arbitrary heterogeneous media [1]. However, the computational cost of WADG grows rapidly with the order of approximation.

…”