We present a high order accurate finite difference method for: a) enforcing nonlinear friction laws, in a consistent and provably stable manner, suitable for efficient explicit time integration; b) dynamic propagation of earthquake ruptures along nonplanar faults; and c) accurate propagation of seismic waves in heterogeneous media with free surface topography.
We solve the first order form of the 3D elastic wave equation on a boundary-conforming curvilinear mesh, in terms of particle velocities and stresses that are collocated in space and time, using summation-by-parts (SBP) finite difference operators in space. Boundary and interface conditions are imposed weakly using penalties. By deriving semi-discrete energy estimates analogous to the continuous energy estimates we prove numerical stability. The finite difference stencils used in this paper are sixth order accurate in the interior and third order accurate close to the boundaries. However, the method is applicable to any spatial operator with a diagonal norm satisfying the SBP property. Time stepping is performed with a 4th order accurate explicit low storage Runge–Kutta scheme, thus yielding a globally fourth order accurate method in both space and time. We show numerical simulations on band limited self-similar fractal faults revealing the complexity of rupture dynamics on rough faults.