Stable and High-Order Accurate Boundary Treatments for the Elastic Wave Equation on Second-Order Form

High-fidelity narrow-stencil finite difference approximations are derived for the elastic wave equation on second-order form and curvilinear grids. The present study focuses on stable numerical boundary treatments for a large class of boundary conditions. A time stable discretization is achieved by combining finite difference operators satisfying a summation-by-parts rule and weak enforcement of boundary conditions, except for Dirichlet boundary conditions which are imposed strongly by injection. The temporal discretization is obtained by an explicit modified equation method. The energy method is used to prove the stability of the semidiscrete and fully discrete problems. This makes precise the convergence properties of the scheme and its usefulness for long time simulations. The accuracy and stability properties of the method are demonstrated for the elastic wave equation in two space dimensions.