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.