Bernstein-Bézier weight-adjusted discontinuous Galerkin methods for wave propagation in heterogeneous media - Guo, Kaihang; Chan, Jesse (Journal of computational physics, 2020)
•A DG scheme for wave propagation in arbitrary heterogeneous media is presented.•Sparse matrix-based implementation of the Bernstein polynomial multiplication.•Representation of the Bernstein polynomial projection matrix in terms of sparse matrices.•Computational complexity is reduced from O(N2d) to O(Nd+1) in d dimensions.•GPU implementation achieves significant speedups with respect to quadrature-base ap-proach. 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. In this work, we propose a Bernstein-Bézier 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 L2 projection, which we carry out using fast Bernstein algorithms. The proposed approach reduces the overall computational complexity from O(N2d) to O(Nd+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.