Wave propagation in anisotropic elastic materials and curvilinear coordinates using a summation-by-parts finite difference method

We develop a fourth order accurate finite di↵erence method for solving the three-dimensional elastic wave equation in general heterogeneous anisotropic materials on curvilinear grids. The proposed method is an extension of the method for isotropic materials, previously described in the paper by Sjögreen and Petersson [J. Sci. Comput. 52 (2012)]. The proposed method discretizes the anisotropic elastic wave equation in second order formulation, using a node centered finite di↵erence method that satisfies the principle of summation by parts. The summation by parts technique results in a provably stable numerical method that is energy conserving. We also generalize and evaluate the super-grid far-field technique for truncating unbounded domains. Unlike the commonly used perfectly matched layers (PML), the super-grid technique is stable for general anisotropic material, because it is based on a coordinate stretching combined with an artificial dissipation. As a result, the discretization satisfies an energy estimate, proving that the numerical approximation is stable. We demonstrate by numerical experiments that, if the super-grid layers are su ciently wide, the errors due to truncating the domain are of the same order as, or smaller than, the propagation errors from the interior of the domain. Applications of the proposed method are demonstrated by three-dimensional simulations of anisotropic wave propagation in crystals.