In this work we develop a high-order method for problems in scalar elastodynamics with nonlinear boundary conditions in a form closely related to those seen in earthquake rupture modeling and other frictional sliding problems. By using summation-by-parts finite difference operators and weak enforcement of boundary conditions with the simultaneous approximation term method, a strictly stable method is developed that dissipates energy at a slightly faster rate than the continuous solution (with the difference in energy dissipation rates vanishing as the mesh is refined). Furthermore, it is shown that unless boundary conditions are formulated in terms of characteristic variables, as opposed to the physical variables in terms of which boundary conditions are more naturally stated, the semi-discretized system of equations can become extremely stiff, preventing efficient solution using explicit time integrators.
These theoretical results are confirmed by several numerical tests demonstrating the high-order convergence rate of the method and the benefits of using strictly stable numerical methods for long time integration.
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