**Rupture Dynamics on Nonplanar Faults with Strongly Rate-Weakening Friction and Off-Fault Plasticity** (poster)

Eric M. Dunham (Stanford)

Natural fault surfaces are nonplanar at all scales. Slip on such faults induces local stress perturbations that lead to irregular rupture propagation and potentially activate off-fault inelastic deformation. Using 2D plane-strain finite difference simulations, we study rupture phenomenology on nonplanar faults in elastic-plastic (Drucker-Prager) media. Motived by high-velocity friction experiments, we use a strongly rate-weakening friction law formulated in a rate-and-state framework. Studies of ruptures on planar faults in elastic media show how strongly rate-weaking friction laws can lead to rupture propagation in the form of slip pulses, provided that the background stress level on the fault is below a critical value. At higher stress levels, ruptures generally take the form of cracks, which can produce several times more slip than slip pulses, even though the differences in dynamic stress drop are minor. This phenomenology holds also for ruptures in elastic-plastic media, though the minimum background stress required for slip pulses to be self-sustaining is increased relative to that in an elastic medium. Plastic deformation generally occurs in the extensional quadrants unless the maximum principal compressive stress is oriented at a low angle to the fault. This is consistent with Templeton and Rice's (JGR, 2008) results with slip-weakening friction laws (which produce crack-like ruptures). However, the extent of plastic deformation (and amount of slip) is reduced when rupture occurs in the slip-pulse instead of crack-like mode. The addition of fault roughness, having amplitude-to-wavelength ratios of 10^{-3} to 10^{-2}, does not greatly change the overall distribution or extent of plastic deformation, but instead modulates its amplitude.This spatial variability of plastic deformation enhances the irregularity of rupture propagation due to fault roughness.

**High-Order Treatment of Fault Boundary Conditions Using Summation-By-Parts Finite Difference Methods** (poster)

Jeremy E. Kozdon (Stanford), Eric M. Dunham (Stanford), Jan Nordström (Uppsala)

High-order numerical methods are ideally suited for earthquake problems, which are primarily limited by available memory rather than CPU time, since they require fewer grid points to achieve the same solution accuracy as low-order methods. Though it is relatively straightforward to apply high-order methods in the interior of the domain, it can be challenging to maintain stability and accuracy near boundaries (e.g., the free surface) and internal interfaces (e.g., faults and layer interfaces). This is particularly problematic for earthquake models since numerical errors near faults degrade the global accuracy of the solution, including ground motion predictions. Despite several efforts to develop high-order fault boundary conditions, no codes have demonstrated greater than second-order accuracy for dynamic rupture problems, even on rate-and-state friction problems with smooth solutions. In this work we use summation-by-parts (SBP) finite difference methods along with a simultaneous approximation term (SAT) to achieve a truly high-order method for dynamic ruptures on faults with rate-and-state friction laws [Carpenter et al., JCP 1999; Nordström & Gustafsson JSC 2003; Nordström SISC 2007]. SBP methods use centered spatial differences in the interior and one-sided differences near the boundary. The transition to one-sided differences is done in a particular manner that permits one to provably maintain stability as well as high-order accuracy. In many methods the boundary conditions are strongly enforced by modifying the difference operator at the boundary so that the solution there exactly satisfies the boundary condition. This approach often results in instability when combined with high-order difference schemes. In contrast, the SAT method enforces the boundary conditions in a weak manner by adding a penalty term to the spatially discretized governing equations. Additional complications arise with rate-and-state friction laws, and several finite difference implementations [Bizzarri et al., GJI, 2001; Rojas et al., GJI, 2009] suffer from extreme stiffness that requires the use of implicit time integration schemes for fields on the fault. This is also the case for the SAT method unless the boundary condition is formulated in terms of characteristic variables (i.e., the combination of stresses and velocities associated with waves entering and exiting the fault). With this formulation, the solution can be advanced using fully explicit time-stepping methods.

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Last updated: September 17, 2009