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

## TL;DR

This paper presents a high-order weight-adjusted discontinuous Galerkin method for 3D poroelastic wave simulations, emphasizing energy stability, micro-heterogeneity handling, and avoiding complex Jacobian diagonalization.

## Contribution

The paper introduces a novel energy-stable penalty flux and a weight-adjusted mass matrix approach for micro-heterogeneity in a high-order DG scheme for poroelastic waves.

## Key findings

- Proven convergence of the numerical scheme.
- High-order accuracy maintained near material discontinuities.
- Comparable or improved performance against spectral element methods.

## Abstract

We introduce a high-order weight-adjusted discontinuous Galerkin (WADG) scheme for the numerical solution of three-dimensional (3D) wave propagation problems in anisotropic porous media. We use a coupled first-order symmetric stress-velocity formulation. Careful attention is directed at (a) the derivation of an energy-stable penalty-based numerical flux, which offers high-order accuracy in presence of material discontinuities, and (b) proper treatment of micro-heterogeneities (sub-element variations) in the numerical scheme. The use of a penalty-based numerical flux avoids the diagonalization of Jacobian matrices into polarized wave constituents necessary when solving element-wise Riemann problems. Micro-heterogeneities are accurately and stably incorporated in the numerical scheme using easily-invertible weight-adjusted mass matrices. The convergence of the proposed numerical scheme is proven and verified by using convergence studies against analytical plane wave solutions. The proposed method is also compared against an existing implementation using the spectral element method to solve the poroelastic wave equation.