[1] The spontaneously propagating shear crack on a frictional interface has proven to be a useful idealization of a natural earthquake. The corresponding boundary value problems are nonlinear and usually require computationally intensive numerical methods for their solution. Assessing the convergence and accuracy of the numerical methods is challenging, as we lack appropriate analytical solutions for comparison. As a complement to other methods of assessment, we compare solutions obtained by two independent numerical methods, a finite difference method and a boundary integral (BI) method. The finite difference implementation, called DFM, uses a traction-at-split-node formulation of the fault discontinuity. The BI implementation employs spectral representation of the stress transfer functional. The three-dimensional (3-D) test problem involves spontaneous rupture spreading on a planar interface governed by linear slip-weakening friction that essentially defines a cohesive law. To get a priori understanding of the spatial resolution that would be required in this and similar problems, we review and combine some simple estimates of the cohesive zone sizes which correspond quite well to the sizes observed in simulations. We have assessed agreement between the methods in terms of the RMS differences in rupture time, final slip, and peak slip rate and related these to median and minimum measures of the cohesive zone resolution observed in the numerical solutions. The BI and DFM methods give virtually indistinguishable solutions to the 3-D spontaneous rupture test problem when their grid spacing Dx is small enough so that the solutions adequately resolve the cohesive zone, with at least three points for BI and at least five node points for DFM. Furthermore, grid-dependent differences in the results, for each of the two methods taken separately, decay as a power law in Dx, with the same convergence rate for each method, the calculations apparently converging to a common, grid interval invariant solution. This result provides strong evidence for the accuracy of both methods. In addition, the specific solution presented here, by virtue of being demonstrably grid-independent and consistent between two very different numerical methods, may prove useful for testing new numerical methods for spontaneous rupture problems.Citation: Day, S. M., L. A. Dalguer, N. Lapusta, and Y. Liu (2005), Comparison of finite difference and boundary integral solutions to three-dimensional spontaneous rupture,

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[1] Although kinematic earthquake source inversions show dominantly pulse-like subshear rupture behavior, seismological observations, laboratory experiments and theoretical models indicate that earthquakes can operate with different rupture styles: either as pulses or cracks, that propagate at subshear or supershear speeds. The determination of rupture style and speed has important implications for ground motions and may inform about the state of stress and strength of active fault zones. We conduct 2D in-plane dynamic rupture simulations with a spectral element method to investigate the diversity of rupture styles on faults governed by velocity-and-state-dependent friction with dramatic velocity-weakening at high slip rate. Our rupture models are governed by uniform initial stresses, and are artificially initiated. We identify the conditions that lead to different rupture styles by investigating the transitions between decaying, steady state and growing pulses, cracks, sub-shear and super-shear ruptures as a function of background stress, nucleation size and characteristic velocity at the onset of severe weakening. Our models show that small changes of background stress or nucleation size may lead to dramatic changes of rupture style. We characterize the asymptotic properties of steady state and self-similar pulses as a function of background stress. We show that an earthquake may not be restricted to a single rupture style, but that complex rupture patterns may emerge that consist of multiple rupture fronts, possibly involving different styles and back-propagating fronts. We also demonstrate the possibility of a super-shear transition for pulse-like ruptures. Finally, we draw connections between our findings and recent seismological observations.

[1] The underestimation of the size of recent megathrust earthquakes illustrates our limited understanding of their spatiotemporal occurrence and governing physics. To unravel their relation to associated subduction dynamics and long-term deformation, we developed a 2-D continuum viscoelastoplastic model that uses an Eulerian-Lagrangian finite difference framework with similar on-and off-fault physics. We extend the validation of this numerical tool to a realistic subduction zone setting that resembles Southern Chile. The resulting quasi-periodic pattern of quasi-characteristic M8-M9 megathrust events compares quantitatively with observed recurrence and earthquake source parameters, albeit at very slow coseismic speeds. Without any data fitting, surface displacements agree with GPS data recorded before and during the 2010 M8.8 Maule earthquake, including the presence of a second-order flexural bulge. These surface displacements show cycle-to-cycle variations of slip deficits, which overall accommodate 5% of permanent internal shortening. We find that thermally (and stress) driven creep governs a spontaneous conditionally stable downdip transition zone between temperatures of 350 ı C and 450 ı C. Ruptures initiate above it (and below the forearc Moho), propagate within it, interspersed by small intermittent events, and arrest below it as ductile shearing relaxes stresses. Ruptures typically propagate upward along lithological boundaries and widen as pressures drop. The main thrust is constrained to be weak due to fluid-induced weakening required to sustain regular subduction and to generate events with natural characteristics (fluid pressures of 75-99% of solid pressures). The agreement with a range of seismological, geodetic, and geological observations demonstrates the validity and strength of this physically consistent seismo-thermo-mechanical approach.

Benchun Duan et al. "A suite of exercises for verifying dynamic earthquake rupture codes. " Seismological Research Letters 89, no. 3 (2018) We describe a set of benchmark exercises that are designed to test if computer codes that simulate dynamic earthquake rupture are working as intended. These types of computer codes are often used to understand how earthquakes operate, and they produce simulation results that include earthquake size, amounts of fault slip, and the patterns of ground shaking and crustal deformation. The benchmark exercises examine a range of features that scientists incorporate in their dynamic earthquake rupture simulations. These include implementations of simple or complex fault geometry, off-fault rock response to an earthquake, stress conditions, and a variety of formulations for fault friction. Many of the benchmarks were designed to investigate scientific problems at the forefronts of earthquake physics and strong ground motions research. The exercises are freely available on our website for use by the scientific community.

[1] Large dynamic stresses near earthquake rupture fronts may induce an inelastic response of the surrounding materials, leading to increased energy absorption that may affect dynamic rupture. We systematically investigate the effects of off-fault plastic energy dissipation in 2-D in-plane dynamic rupture simulations under velocity-and-statedependent friction with severe weakening at high slip velocity. We find that plasticity does not alter the nature of the transitions between different rupture styles (decaying versus growing, pulse-like versus crack-like, and subshear versus supershear ruptures) but increases their required background stress and nucleation size. We systematically quantify the effect of amplitude and orientation of background shear stresses on the asymptotic properties of self-similar pulse-like ruptures: peak slip rate, rupture speed, healing front speed, slip gradient, and the relative contribution of plastic strain to seismic moment. Peak slip velocity and rupture speed remain bounded. From fracture mechanics arguments, we derive a nonlinear relation between their limiting values, appropriate also for crack-like and supershear ruptures. At low background stress, plasticity turns self-similar pulses into steady state pulses, for which plastic strain contributes significantly to the seismic moment. We find that the closeness to failure of the background stress state is an adequate predictor of rupture speed for relatively slow events. Our proposed relations between state of stress and earthquake source properties in the presence of off-fault plasticity may contribute to the improved interpretation of earthquake observations and to pseudodynamic source modeling for ground motion prediction.

[1] The physics governing the seismic cycle at seismically active subduction zones remains poorly understood due to restricted direct observations in time and space. To investigate subduction zone dynamics and associated interplate seismicity, we validate a continuum, visco-elasto-plastic numerical model with a new laboratory approach (Paper 1). The analogous laboratory setup includes a visco-elastic gelatin wedge underthrusted by a rigid plate with defined velocity-weakening and -strengthening regions. Our geodynamic simulation approach includes velocity-weakening friction to spontaneously generate a series of fast frictional instabilities that correspond to analog earthquakes. A match between numerical and laboratory source parameters is obtained when velocity-strengthening is applied in the aseismic regions to stabilize the rupture. Spontaneous evolution of absolute stresses leads to nucleation by coalescence of neighboring patches, mainly occurring at evolving asperities near the seismogenic zone limits. Consequently, a crack-, or occasionally even pulse-like, rupture propagates toward the opposite side of the seismogenic zone by increasing stresses ahead of its rupture front, until it arrests on a barrier. The resulting surface displacements qualitatively agree with geodetic observations and show landward and, from near the downdip limit, upward interseismic motions. These are rebound and reversed coseismically. This slip increases adjacent stresses, which are relaxed postseismically by afterslip and thereby produce persistent seaward motions. The wide range of observed physical phenomena, including back-propagation and repeated slip, and the agreement with laboratory results demonstrate that visco-elasto-plastic geodynamic models with rate-dependent friction form a new tool that can greatly contribute to our understanding of the seismic cycle at subduction zones.Citation: van Dinther, Y., T. V. Gerya, L. A. Dalguer, F. Corbi, F. Funiciello, and P. M. Mai (2012), The seismic cycle at subduction thrusts: 2. Dynamic implications of geodynamic simulations validated with laboratory models, J. Geophys.

[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 staggeredgrid 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 faultdiscontinuity 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.Citation: Dalguer, L. A., and S. M. Day (2007), Staggered-grid split-node method for spontaneous rupture simulation, J. Geophys.