Dual-pairing summation by parts finite difference methods for large scale elastic wave simulations in 3D complex geometries

Abstract

High-order accurate summation-by-parts (SBP) finite difference (FD) methods constitute efficient numerical methods for simulating large-scale hyperbolic wave propagation problems. Traditional SBP FD operators that approximate first-order spatial derivatives with central-difference stencils often have spurious unresolved numerical wave-modes in their computed solutions. Recently derived high order accurate dual pair SBP operators based non-central (upwind) FD stencils have the potential to suppress these poisonous spurious wave-modes on marginally resolved computational grids. In this paper, we demonstrate that not all high order dual pair SBP FD operators are applicable. Numerical dispersion relation analysis shows that odd-order dual pair SBP FD operators also support spurious unresolved high-frequency wave-modes on marginally resolved meshes. Meanwhile, even-order dual pair SBP FD operators (of order 2,4,6) do not support spurious unresolved high frequency wave-modes and also have better numerical dispersion properties.For all the dual pair SBP FD operators we discretise the three space dimensional (3D) elastic wave equation on boundary-conforming curvilinear meshes. Using the energy method we prove that the semi-discrete approximation is stable and energy-conserving. We derive a priori error estimate and prove the convergence of the numerical error for smooth solutions. Numerical experiments for the 3D elastic wave equation in complex geometries corroborate the theoretical analysis. Numerical simulations of the 3D elastic wave equation in heterogeneous media with complex non-planar free surface topography are given, including numerical simulations of community developed seismological benchmark problems. Computational results show that even-order dual pair SBP FD operators are more efficient, robust and less prone to numerical dispersion errors on marginally resolved meshes when compared to the odd-order dual pair and traditional SBP FD operators. Finally, scaling tests demonstrate nearly perfect strong scaling and verify the efficiency of our parallel implementation of the high order upwind SBP methods.