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Simulation of Earthquake Rupture Dynamics in Complex Geometries Using Coupled Finite Difference and Finite Volume Methods

Published online by Cambridge University Press:  22 January 2015

Ossian O'Reilly ,
Jan Nordström ,
Jeremy E. Kozdon  and
Eric M. Dunham
Ossian O'Reilly*
Affiliation:
Department of Geophysics, Stanford University, CA 94305-2215, USA Department of Mathematics, Division of Computational Mathematics, Linköping University, SE-581 83 Linköping, Sweden
Jan Nordström
Affiliation:
Department of Mathematics, Division of Computational Mathematics, Linköping University, SE-581 83 Linköping, Sweden
Jeremy E. Kozdon
Affiliation:
Department of Applied Mathematics, Naval Postgraduate School, Monterey, CA 93943-5216, USA
Eric M. Dunham
Affiliation:
Department of Geophysics, Stanford University, CA 94305-2215, USA Institute for Computational and Mathematical Engineering, Stanford University, CA 94305-4042, USA
*
*Email addresses: ooreilly@stanford.edu (O. O'Reilly), jan.nordstrom@liu.se (J. Nordström), jekozdon@nps.edu (J. E. Kozdon), edunham@stanford.edu (E. M. Dunham)
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Abstract

We couple a node-centered finite volume method to a high order finite difference method to simulate dynamic earthquake ruptures along nonplanar faults in two dimensions. The finite volume method is implemented on an unstructured mesh, providing the ability to handle complex geometries. The geometric complexities are limited to a small portion of the overall domain and elsewhere the high order finite difference method is used, enhancing efficiency. Both the finite volume and finite difference methods are in summation-by-parts form. Interface conditions coupling the numerical solution across physical interfaces like faults, and computational ones between structured and unstructured meshes, are enforced weakly using the simultaneous-approximation-term technique. The fault interface condition, or friction law, provides a nonlinear relation between fields on the two sides of the fault, and allows for the particle velocity field to be discontinuous across it. Stability is proved by deriving energy estimates; stability, accuracy, and efficiency of the hybrid method are confirmed with several computational experiments. The capabilities of the method are demonstrated by simulating an earthquake rupture propagating along the margins of a volcanic plug.

Keywords

35L0535L6535Q3565M0665M0865M1265Z05Elastic wavesearthquakehigh order finite difference finite volumesummation-by-partssimultaneous approximation termnonlinear boundary conditions
Type
Research Article
Information
Communications in Computational Physics , Volume 17 , Issue 2 , February 2015 , pp. 337 - 370
Copyright
Copyright © Global Science Press Limited 2015 

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References

[1] Aagaard, B.T., Finite-element simulations of earthquakes, Ph.D. thesis, 1999.Google Scholar
[2] Amsallem, D., Nordström, J., High-order accurate difference schemes for the hodgkin-huxley equations, J. Comput. Phys. 252 (2013) 573–590.Google Scholar
[3] Andrews, D., Rupture propagation with finite stress in antiplane strain, J. Geophys. Res. 81 (1976) 3575–3582.Google Scholar
[4] Aochi, H., Fukuyama, E., Three-dimensional nonplanar simulation of the 1992 Landers earthquake, J. Geophys. Res. 107 (2002) 2001.Google Scholar
[5] Appelö, D., Hagstrom, T., Kreiss, G., Perfectly matched layers for hyperbolic systems: general formulation, well-posedness, and stability, SIAM J. Appl. Math. (2006) 1–23.CrossRefGoogle Scholar
[6] Badea, L., Ionescu, I.R., Wolf, S. , Schwarz method for earthquake source dynamics, J. Comput. Phys. 227 (2008) 3824–3848.Google Scholar
[7] Barall, M., A grid-doubling finite-element technique for calculating dynamic three-dimensional spontaneous rupture on an earthquake fault, Geophysical Journal International 178 (2009) 845–859.CrossRefGoogle Scholar
[8] Benjemaa, M., Glinsky-Olivier, N., Cruz-Atienza, V., Virieux, J., 3-D dynamic rupture simulations by a finite volume method, Geophys. J. Int. 178 (2009) 541–560.Google Scholar
[9] Benjemaa, M., Glinsky-Olivier, N., Cruz-Atienza, V., Virieux, J., Piperno, S., Dynamic non-planar crack rupture by a finite volume method, Geophys. J. Int. 171 (2007) 271–285.Google Scholar
[10] Carpenter, M., Gottlieb, D., Abarbanel, S., Time-stable boundary conditions for finite-difference schemes solving hyperbolic systems: methodology and application to high-order compact schemes, J. Comput. Phys. (1993).Google Scholar
[11] Carpenter, M., Kennedy, C., Fourth-order 2N-storage Runge-Kutta schemes, Nasa Technical Memorandum 109112 (1994).Google Scholar
[12] Carpenter, M., Nordström, J., Gottlieb, D., A stable and conservative interface treatment of arbitrary spatial accuracy, J. Comput. Phys. 148 (1999) 341–365.Google Scholar
[13] Carpenter, M.H., Nordström, J., Gottlieb, D., Revisiting and extending interface penalties for multi-domain summation-by-parts operators, J. Sci. Comput. 45 (2010) 118–150.Google Scholar
[14] Cruz-Atienza, V., Virieux, J., Dynamic rupture simulation of non-planar faults with a finite-difference approach, Geophys. J. Int. 158 (2004) 939–954.Google Scholar
[15] Cruz-Atienza, V., Virieux, J., Aochi, H., 3D finite-difference dynamic-rupture modeling along nonplanar faults, Geophys. 72 (2007) SM123–SM137.Google Scholar
[16] Day, S., Three-dimensional finite difference simulation of fault dynamics: rectangular faults with fixed rupture velocity, Bull. Seismol. Soc. Am. 72 (1982) 705–727.Google Scholar
[17] De La Puente, J., Ampuero, J., Käser, M., Dynamic rupture modeling on unstructured meshes using a discontinuous Galerkin method, J. geophys. Res 114 (2009) B10302.Google Scholar
[18] Duan, B., Oglesby, D., Nonuniform prestress from prior earthquakes and the effect on dynamics of branched fault systems, J. Geophys. Res 112 (2007) B05308.Google Scholar
[19] Ely, G.P., Day, S.M., Minster, J.B., A support-operator method for viscoelastic wave modelling in 3-D heterogeneous media, Geophys. J. Int. 172 (2008) 331–344.Google Scholar
[20] Festa, G., Vilotte, J.P., The Newmark scheme as velocity-stress time-staggering: an efficient PML implementation for spectral element simulations of elastodynamics, Geophys. J. Int. 161 (2005) 789–812.CrossRefGoogle Scholar
[21] Gassner, G.J., A skew-symmetric discontinuous galerkin spectral element discretization and its relation to sbp-sat finite difference methods, SIAM Journal on Scientific Computing 35 (2013) A1233–A1253.Google Scholar
[22] Geubelle, P.H., Rice, J.R., A spectral method for three-dimensional elastodynamic fracture problems, Journal of the Mechanics and Physics of Solids 43 (1995) 1791–1824.Google Scholar
[23] Gong, J., Nordström, J., A stable and efficient hybrid scheme for viscous problems in complex geometries, J. Comput. Phys. 226 (2007) 1291–1309.CrossRefGoogle Scholar
[24] Gong, J., Nordström, J., Interface procedures for finite difference approximations of the advection-diffusion equation, J. Comput. Appl. Math. 236 (2011) 602–620.CrossRefGoogle Scholar
[25] Gustafsson, B., The convergence rate for difference approximations to mixed initial boundary value problems, Math. Comp 29 (1975) 396–406.CrossRefGoogle Scholar
[26] Gustafsson, B., Kreiss, H., Oliger, J., Time dependent problems and difference methods, volume 24, Wiley-Interscience, 1995.Google Scholar
[27] Hagstrom, T., Mar-Or, A., Givoli, D., High-order local absorbing conditions for the wave equation: Extensions and improvements, J. Comput. Phys. 227 (2008) 3322–3357.Google Scholar
[28] Harris, R., Barall, M., Archuleta, R., Dunham, E., Aagaard, B., Ampuero, J., Bhat, H., Cruz-Atienza, V., Dalguer, L., Dawson, P., et al., The SCEC/USGS dynamic earthquake rupture code verification exercise, Seismol. Res. Lett. 80 (2009) 119–126.Google Scholar
[29] Hesthaven, J.S., Warburton, T., Nodal discontinuous Galerkin methods: algorithms, analysis, and applications, volume 54, Springer, 2007.Google Scholar
[30] Igel, H., Käser, M., Stupazzini, M., Seismic wave propagation in media with complex geometries, simulation of., 2009.Google Scholar
[31] Iverson, R., Dzurisin, D., Gardner, C., Gerlach, T., LaHusen, R., Lisowski, M., Major, J., Malone, S., Messerich, J., Moran, S., et al., Dynamics of seismogenic volcanic extrusion at Mount St Helens in 2004-05, Nat. 444 (2006) 439–443.Google Scholar
[32] Kame, N., Yamashita, T., Simulation of the spontaneous growth of a dynamic crack without constraints on the crack tip path, Geophys. J. Int. 139 (1999) 345–358.CrossRefGoogle Scholar
[33] Kaneko, Y., Lapusta, N., Ampuero, J.P., Spectral element modeling of spontaneous earthquake rupture on rate and state faults: Effect of velocity-strengthening friction at shallow depths, J. Geophys. Res. 113 (2008) B09317.Google Scholar
[34] Käser, M., Mai, P., Dumbser, M., Accurate calculation of fault-rupture models using the high- order discontinuous Galerkin method on tetrahedral meshes, Bull. Seismol. Soc. Am. 97 (2007) 1570–1586.Google Scholar
[35] Kozdon, J.E., Dunham, E.M., Nordström, J., Interaction of waves with frictional interfaces using summation-by-parts difference operators: Weak enforcement of nonlinear boundary conditions, J. Sci. Comput. 50 (2012) 341–367.CrossRefGoogle Scholar
[36] Kozdon, J.E., Dunham, E.M., Nordström, J., Simulation of dynamic earthquake ruptures in complex geometries using high-order finite difference methods, J. Sci. Comput. 55 (2013) 92–124.Google Scholar
[37] Kreiss, H., Initial boundary value problems for hyperbolic systems, Commun. Pure Appl. Math. 23 (1970) 277–298.Google Scholar
[38] Kreiss, H., Scherer, G., Finite element and finite difference methods for hyperbolic partial differential equations, Academic Press, 1974.Google Scholar
[39] Kreiss, H., Scherer, G., On the existence of energy estimates for difference approximations for hyperbolic systems, Tech. Rep. (1977).Google Scholar
[40] Ma, S., Archuleta, R., Finite element modeling of earthquake dynamic rupture on 2D nonpla- nar dipping faults, in: AGU Fall Meeting Abstracts, volume 1, p. 0938.Google Scholar
[41] Ma, S., Archuleta, R.J., Liu, P., Hybrid modeling of elastic p-sv wave motion: a combined finite-element and staggered-grid finite-difference approach, Bull. Seismol. Soc. Am. 94 (2004) 1557–1563.Google Scholar
[42] Madariaga, R., Olsen, K., Archuleta, R., Modeling dynamic rupture in a 3D earthquake fault model, Bull. Seismol. Soc. Am. 88 (1998) 1182–1197.Google Scholar
[43] Mattsson, K., Nordström, J., High order finite difference methods for wave propagation in discontinuous media, J. Comput. Phys. 220 (2006) 249–269.CrossRefGoogle Scholar
[44] Moczo, P., Kristek, J., Galis, M., Pazak, P., Balazovjech, M., The finite-difference and finite-element modeling of seismic wave propagation and earthquake motion, Acta Phys. Slovaca 57 (2007).Google Scholar
[45] Moran, S., Malone, S., Qamar, A., Thelen, W., Wright, A., Caplan-Auerbach, J., Seismicity associated with renewed dome building at Mount St. Helens, 2004-2005, Prof. Pap. U.S Geol. Surv. (2008) 27–60.Google Scholar
[46] Nordström, J., Error bounded schemes for time-dependent hyperbolic problems, SIAM J. Sci. Comput. 30 (2007) 46–59.Google Scholar
[47] Nordström, J., Linear and nonlinear boundary conditions for wave propagation problems, in: Recent Developments in the Numerics of Nonlinear Hyperbolic Conservation Laws, Springer, 2013, pp. 283–299.CrossRefGoogle Scholar
[48] Nordström, J., Forsberg, K., Adamsson, C., Eliasson, P., Finite volume methods, unstructured meshes and strict stability for hyperbolic problems, Appl. Numer. Math. 45 (2003) 453–473.Google Scholar
[49] Nordström, J., Gong, J., A stable hybrid method for hyperbolic problems, J. Comput. Phys. 212 (2006) 436–453.CrossRefGoogle Scholar
[50] Nordström, J., Gong, J., Van der Weide, E., Svärd, M., A stable and conservative high order multi-block method for the compressible Navier-Stokes equations, J. Comput. Phys. 228 (2009a) 9020–9035.Google Scholar
[51] Nordström, J., Gustafsson, R., High order finite difference approximations of electromagnetic wave propagation close to material discontinuities, J. Sci. Comput. 18 (2003) 215–234.Google Scholar
[52] Nordström, J., Ham, F., Shoeybi, M., Van der Weide, E., Svärd, M., Mattsson, K., Iaccarino, G., Gong, J., A hybrid method for unsteady inviscid fluid flow, Comput. Fluids 38 (2009b) 875–882.Google Scholar
[53] Nordström, J., Mattsson, K., Swanson, C., Boundary conditions for a divergence free velocity-pressure formulation of the Navier-Stokes equations, J. Comput. Phys. 225 (2007) 874–890.CrossRefGoogle Scholar
[54] Olsson, P., Summation by parts, projections, and stability. I, Math. Comput. 64 (1995) 1035–1035.Google Scholar
[55] Pelties, C., de la Puente, J., Ampuero, J.P., Brietzke, G.B., Kser, M., Three-dimensional dynamic rupture simulation with a high-order discontinuous Galerkin method on unstructured tetra-hedral meshes, J. Geophys. Res. 117 (2012) B02309.Google Scholar
[56] Rice, J., Lapusta, N., Ranjith, K., Rate and state dependent friction and the stability of sliding between elastically deformable solids, J. Mech. Phys. Solids 49 (2001) 1865–1898.CrossRefGoogle Scholar
[57] Roache, P.J., Verification and validation in computational science and engineering, Hermosa Publishers, 1998.Google Scholar
[58] Ruina, A., Stability of steady frictional slipping, J. Appl. Mech. 50 (1983) 343–349.Google Scholar
[59] Strand, B., Summation by parts for finite difference approximations for d/dx, J. Comput. Phys. 110 (1994) 47–67.Google Scholar
[60] Svärd, M., Carpenter, M.H., Nordström, J., A stable high-order finite difference scheme for the compressible Navier-Stokes equations, far-field boundary conditions, J. Comput. Phys. 225 (2007) 1020–1038.Google Scholar
[61] Svärd, M., Nordström, J., On the order of accuracy for difference approximations of initial boundary value problems, J. Comput. Phys. 218 (2006) 333–352.Google Scholar
[62] Svärd, M., Nordström, J., A stable high-order finite difference scheme for the compressible Navier-Stokes equations: No-slip wall boundary conditions, J. Comput. Phys. 227 (2008) 4805–4824.Google Scholar
[63] Svärd, M., Nordström, J., Review of summation-by-parts schemes for initialboundary-value problems, J. Comput. Phys. 268 (2014) 17–38.Google Scholar
[64] Tada, T., Fukuyama, E., Madariaga, R., Non-hypersingular boundary integral equations for 3-D non-planar crack dynamics, Comput. Mech. 25 (2000) 613–626.Google Scholar
[65] Tago, J., Cruz-Atienza, V., Virieux, J., Etienne, V., Sánchez-Sesma, F., A 3D hp-adaptive discontinuous Galerkin method for modeling earthquake dynamics, J. Geophys. Res. 117 (2012) B09312.Google Scholar
[66] Templeton, E., Baudet, A., Bhat, H., Dmowska, R., Rice, J., Rosakis, A., Rousseau, C., Finite element simulations of dynamic shear rupture experiments and dynamic path selection along kinked and branched faults, J. Geophys. Res. 114 (2009) B08304.Google Scholar
[67] Templeton, E., Bhat, H., Dmowska, R., Rice, J., Dynamic rupture through a branched fault configuration at Yucca Mountain, and resulting ground motions, Bull. Seismol. Soc. Am. 100 (2010) 1485–1497.Google Scholar
[68] Thompson, J.F., Soni, B.K., Weatherill, N.P., Handbook of Grid Generations, CRC press, 1999.Google Scholar
[69] Virieux, J., Madariaga, R., Dynamic faulting studied by a finite difference method, Bull. Seis- mol. Soc. Am. 72 (1982) 345–369.Google Scholar
[70] Zhang, H., Chen, X., Dynamic rupture on a planar fault in three-dimensional half spacei. theory, Geophys. J. Internat. 164 (2006) 633–652.Google Scholar