Problems in elastodynamics with nonlinear boundary conditions, such as those arising when modeling earthquake rupture propagation along internal interfaces (faults) governed by nonlinear friction laws, are inherently boundary driven. For such problems, stable and accurate enforcement of boundary conditions is essential for obtaining globally accurate numerical solutions (and predictions of ground motion in earthquake simulations). High-order finite difference methods are a natural choice for problems like these involving wave propagation, but enforcement of boundary conditions is complicated by the fact that the stencil must transition to one-sided near the boundary.
In this work we develop a high-order method for tensor elasticity with faults whose strength is a nonlinear function of sliding velocity and a set of internal state variables obeying differential evolution equations (a mathematical framework known as rate-and-state friction). The method is based on summation-by-parts finite difference operators and weak enforcement of boundary conditions using the simultaneous approximation term method. We prove that the method is strictly stable and dissipates energy at a slightly faster rate than the continuous solution (with the difference in energy dissipation rates vanishing as the mesh is refined).
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