Staggered-grid split-node method for spontaneous rupture simulation

[1] We adapt the traction-at-split-node method for spontaneous rupture simulations to the velocity-stress staggered-grid finite difference scheme. The staggered-grid implementation introduces both velocity and stress discontinuities via split nodes. The staggered traction components on the fault plane are interpolated to form the traction vector at split nodes, facilitating alignment of the vectors of sliding friction and slip velocity. To simplify the split-node partitioning of the equations of motion, spatial differencing is reduced from fourth to second order along the fault plane, but in the remainder of the grid the spatial differencing scheme remains identical to conventional spatially fourth-order three-dimensional staggered-grid schemes. The resulting staggered-grid split node (SGSN) method has convergence rates relative to rupture-time, final-slip, and peak-slip-velocity metrics that are very similar to the corresponding rates for both a partly staggered split-node code (DFM) and the boundary integral method. The SGSN method gives very accurate solutions ( in the sense that errors are comparable to the uncertainties in the reference solution) when the median resolution of the cohesive zone is 4.4 grid points. Combined with previous results for other grid types and other fault-discontinuity approximations, the SGSN results demonstrate that accuracy in finite difference solutions to the spontaneous rupture problem is controlled principally by the scheme used to represent the fault discontinuity, and is relatively insensitive to the grid geometry used to represent the continuum. The method provides an efficient and accurate means of adding spontaneous rupture capability to velocity-stress staggered-grid finite difference codes, while retaining the computational advantages of those codes for problems of wave propagation in complex media.