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

TL;DR

This paper introduces a Bernstein-Bézier weight-adjusted discontinuous Galerkin method that efficiently simulates wave propagation in heterogeneous media, significantly reducing computational costs while maintaining high accuracy.

Contribution

The paper develops a novel BBWADG method that approximates sub-cell heterogeneities with polynomials and employs fast Bernstein algorithms to lower computational complexity.

Findings

Achieves high-order accuracy in heterogeneous media simulations.

Reduces computational complexity from O(N^{2d}) to O(N^{d+1}) in d dimensions.

GPU implementation confirms theoretical efficiency gains.

Abstract

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. However, the computational cost of WADG grows rapidly with the order of approximation. In this work, we propose a Bernstein-B\'ezier weight-adjusted discontinuous Galerkin method (BBWADG) to address this cost. By approximating sub-cell heterogeneities by a fixed degree polynomial, the main steps of WADG can be expressed as polynomial multiplication and $L^2$ projection, which we carry out using fast Bernstein algorithms. The proposed approach reduces the overall computational complexity from $O(N^{2d})$ to $O(N^{d+1})$ in $d$ dimensions. Numerical experiments illustrate the accuracy of the proposed approach, and computational experiments for a GPU implementation of BBWADG verify that this theoretical complexity is achieved in practice.