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Description

We describe a new stable finite difference method for the elastic wave equation in second order form on complex domains with stress free boundary conditions. Finite difference discretizations of the elastic wave equation in second order form are highly efficient and easy to implement (compared to stag- gered discretizations). Due to these favorable properties the methods were popular in the early days of seismic modeling. However, it was soon discovered that the early methods suffered from two main flaws: The lack of stable discretization of the free surface boundary condition at high ratios between the P and S-wave speed, and the handling of complex geometries needed for realistic topography. Recently a remedy to the free surface instabilities was presented in a paper by Nilsson, Petersson, Sjogreen and Kreiss. They introduced a summation-by-parts discretization were, at a free surface, one sided difference operators where used for mixed derivatives in the equations. For their discretization they were able to derive a guaranteed stable discretization of the stress free boundary condition, even when the material properties vary rapidly on the grid. The method was proved to be second order accurate and stable for all ratios of the speeds of the P and S-wave. Here we generalize the results of Nilsson et al. to curvilinear grids. Using summation by parts techniques we show that there exist a corresponding discretization of the free surface boundary condition on a curvilinear grid. We prove that the discretization is stable and energy conserving both in semi-discrete and fully discrete form. We also establish that the method is second order accurate. Finally, we present computations on overset grids, where we combine the new curvilinear and the Cartesian discretization into an efficient and accurate numerical method for simulations of elastic waves on complex domains.