Wavefield Separation Algorithm of Helmholtz Theory-Based Discontinuous Galerkin Method Using Unstructured Meshes on GPU

Abstract

In the field of geophysics, the Helmholtz decomposition (HD) formula is mainly numerically discretized by the finite difference method (FDM), which limits its application to a uniform regular grid only. Few scholars note wavefield separation on unstructured grids. In this study, we aim to develop a wavefield separation algorithm to separate P- and S-wavefields on nonuniform grids. Our scheme is based on an isotropic elastic wave equation. We first transform the HD formula into a weak integral form using the discontinuous Galerkin method (DGM). Then, we consider two types of unstructured meshes—triangle and quadrangle, which are more suitable for complex structures. Moreover, to reduce time costs, the single graphic processor unit (GPU) device is used to improve the computational efficiency. We perform a unified DGM operator by transforming unstructured triangles and quadrangles into standard reference elements using coordinate transformation. Our proposed HD operator enables us to effectively separate P- and S-wavefields on unstructured meshes. We carry out the wavefield separation simulation and calculate the numerical solutions in the homogeneous carbonate model, Graben model, and SEG/EAGE model. The homogeneous model verifies the correctness, availability, and superiority of our proposed separated operator, and the other numerical results show excellent performance for P/S-wavefield separation on unstructured meshes.