Time dependent boundary conditions for hyperbolic systems
Journal of Computational Physics
Time-dependent boundary conditions for hyperbolic systems, II
Journal of Computational Physics
Summation by parts for finite difference approximations for d/dx
Journal of Computational Physics
Summation by parts, projections, and stability. II
Mathematics of Computation
A stable and conservative interface treatment of arbitrary spatial accuracy
Journal of Computational Physics
High-order finite difference methods, multidimensional linear problems, and curvilinear coordinates
Journal of Computational Physics
Boundary Procedures for Summation-by-Parts Operators
Journal of Scientific Computing
Journal of Scientific Computing
Finite volume methods, unstructured meshes and strict stability for hyperbolic problems
Applied Numerical Mathematics
Summation by parts operators for finite difference approximations of second derivatives
Journal of Computational Physics
Journal of Scientific Computing
On the order of accuracy for difference approximations of initial-boundary value problems
Journal of Computational Physics
Journal of Computational Physics
High-order local absorbing conditions for the wave equation: Extensions and improvements
Journal of Computational Physics
Error Bounded Schemes for Time-dependent Hyperbolic Problems
SIAM Journal on Scientific Computing
Stable Boundary Treatment for the Wave Equation on Second-Order Form
Journal of Scientific Computing
Journal of Computational Physics
Journal of Scientific Computing
Journal of Computational Physics
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We present a high-order difference method for problems in elastodynamics involving the interaction of waves with highly nonlinear frictional interfaces. We restrict our attention to two-dimensional antiplane problems involving deformation in only one direction. Jump conditions that relate tractions on the interface, or fault, to the relative sliding velocity across it are of a form closely related to those used in earthquake rupture models and other frictional sliding problems. By using summation-by-parts (SBP) finite difference operators and weak enforcement of boundary and interface conditions, a strictly stable method is developed. Furthermore, it is shown that unless the nonlinear interface conditions are formulated in terms of characteristic variables, as opposed to the physical variables in terms of which they are more naturally stated, the semi-discretized system of equations can become extremely stiff, preventing efficient solution using explicit time integrators.The use of SBP operators also provides a rigorously defined energy balance for the discretized problem that, as the mesh is refined, approaches the exact energy balance in the continuous problem. This enables one to investigate earthquake energetics, for example the efficiency with which elastic strain energy released during rupture is converted to radiated energy carried by seismic waves, rather than dissipated by frictional sliding of the fault. These theoretical results are confirmed by several numerical tests in both one and two dimensions demonstrating the computational efficiency, the high-order convergence rate of the method, the benefits of using strictly stable numerical methods for long time integration, and the accuracy of the energy balance.