Skip to main content
SpringerLink
Log in

Simulation of Dynamic Earthquake Ruptures in Complex Geometries Using High-Order Finite Difference Methods

  • Published:
Journal of Scientific Computing Aims and scope Submit manuscript

Abstract

We develop a stable and high-order accurate finite difference method for problems in earthquake rupture dynamics in complex geometries with multiple faults. The bulk material is an isotropic elastic solid cut by pre-existing fault interfaces that accommodate relative motion of the material on the two sides. The fields across the interfaces are related through friction laws which depend on the sliding velocity, tractions acting on the interface, and state variables which evolve according to ordinary differential equations involving local fields.

The method is based on summation-by-parts finite difference operators with irregular geometries handled through coordinate transforms and multi-block meshes. Boundary conditions as well as block interface conditions (whether frictional or otherwise) are enforced weakly through the simultaneous approximation term method, resulting in a provably stable discretization.

The theoretical accuracy and stability results are confirmed with the method of manufactured solutions. The practical benefits of the new methodology are illustrated in a simulation of a subduction zone megathrust earthquake, a challenging application problem involving complex free-surface topography, nonplanar faults, and varying material properties.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7
Fig. 8
Fig. 9
Fig. 10

Similar content being viewed by others

References

  1. Aagaard, B.T., Heaton, T.H., Hall, J.F.: Dynamic earthquake ruptures in the presence of lithostatic normal stresses: implications for friction models and heat production. Bull. Seismol. Soc. Am. 91(6), 1765–1796 (2001). doi:10.1785/0120000257

    Article  Google Scholar 

  2. Ampuero, J.-P.: Etude physique et numérique de la nucléation des séismes. PhD thesis, Univ. Denis Diderot, Paris (2002)

  3. Andrews, D.J.: Rupture propagation with finite stress in antiplane strain. J. Geophys. Res. 81(20), 3575–3582 (1976). doi:10.1029/JB081i020p03575

    Article  Google Scholar 

  4. Andrews, D.J.: Dynamic plane-strain shear rupture with a slip-weakening friction law calculated by a boundary integral method. Bull. Seismol. Soc. Am. 75(1), 1–21 (1985)

    Google Scholar 

  5. Aochi, H., Fukuyama, E., Matsu’ura, M.: Selectivity of spontaneous rupture propagation on a branched fault. Geophys. Res. Lett. 27(22), 3635–3638 (2000). doi:10.1029/2000GL011560

    Article  Google Scholar 

  6. Appelö, D., Petersson, N.A.: A stable finite difference method for the elastic wave equation on complex geometries with free surfaces. Commun. Comput. Phys. 5(1), 84–107 (2009)

    MathSciNet  Google Scholar 

  7. Appelö, D., Hagstrom, T., Kreiss, G.: Perfectly matched layers for hyperbolic systems: general formulation, well-posedness, and stability. SIAM J. Appl. Math. 67(1), 1–23 (2006). doi:10.1137/050639107

    Article  MathSciNet  MATH  Google Scholar 

  8. Bayliss, A., Jordan, K.E., Lemesurier, B.J., Turkel, E.: A fourth-order accurate finite-difference scheme for the computation of elastic waves. Bull. Seismol. Soc. Am. 76(4), 1115–1132 (1986)

    Google Scholar 

  9. Carpenter, M.H., Kennedy, C.A.: Fourth-order 2N-storage Runge-Kutta schemes. Technical report NASA TM-109112, National Aeronautics and Space Administration, Langley Research Center, Hampton, VA (1994)

  10. Carpenter, M.H., Nordström, J., Gottlieb, D.: A stable and conservative interface treatment of arbitrary spatial accuracy. J. Comput. Phys. 148(2), 341–365 (1999). doi:10.1006/jcph.1998.6114

    Article  MathSciNet  MATH  Google Scholar 

  11. Cruz-Atienza, V.M., Virieux, J.: Dynamic rupture simulation of non-planar faults with a finite-difference approach. Geophys. J. Int. 158(3), 939–954 (2004). doi:10.1111/j.1365-246X.2004.02291.x

    Article  Google Scholar 

  12. Das, S.: A numerical method for determination of source time functions for general three-dimensional rupture propagation. Geophys. J. R. Astron. Soc. 62(3), 591–604 (1980). doi:10.1111/j.1365-246X.1980.tb02593.x

    Article  MATH  Google Scholar 

  13. Das, S., Kostrov, B.V.: An investigation of the complexity of the earthquake source time function using dynamic faulting models. J. Geophys. Res. 93(B7), 8035–8050 (1988). doi:10.1029/JB093iB07p08035

    Article  Google Scholar 

  14. Day, S.M.: Three-dimensional finite difference simulation of fault dynamics: rectangular faults with fixed rupture velocity. Bull. Seismol. Soc. Am. 72(3), 705–727 (1982)

    Google Scholar 

  15. Day, S.M., Dalguer, L.A., Lapusta, N., Liu, Y.: Comparison of finite difference and boundary integral solutions to three-dimensional spontaneous rupture. J. Geophys. Res. 110, B12307 (2005). doi:10.1029/2005JB003813

    Article  Google Scholar 

  16. de la Puente, J., Ampuero, J.-P., Käser, M.: Dynamic rupture modeling on unstructured meshes using a discontinuous Galerkin method. J. Geophys. Res. 114, B10302 (2009). doi:10.1029/2008JB006271

    Article  Google Scholar 

  17. Dunham, E.M., Belanger, D., Cong, L., Kozdon, J.E.: Earthquake ruptures with strongly rate-weakening friction and off-fault plasticity, Part 1: Planar faults. Bull. Seismol. Soc. Am. 101(5), 2296–2307 (2011). doi:10.1785/0120100075

    Article  Google Scholar 

  18. Dunham, E.M., Belanger, D., Cong, L., Kozdon, J.E.: Earthquake ruptures with strongly rate-weakening friction and off-fault plasticity, Part 2: Nonplanar faults. Bull. Seismol. Soc. Am. 101(5), 2308–2322 (2011). doi:10.1785/0120100076

    Article  Google Scholar 

  19. Festa, G., Vilotte, J.-P.: The Newmark scheme as velocity and stress time-staggering: an efficient PML implementation for spectral element simulations of elastodynamics. Geophys. J. Int. 161(3), 789–812 (2005). doi:10.1111/j.1365-246X.2005.02601.x

    Article  Google Scholar 

  20. Fornberg, B.: The pseudospectral method: accurate representation of interfaces in elastic wave calculations. Geophysics 53(5), 625–637 (1988). doi:10.1190/1.1442497

    Article  Google Scholar 

  21. Geubelle, P.H., Rice, J.R.: A spectral method for three-dimensional elastodynamic fracture problems. J. Mech. Phys. Solids 43(11), 1791–1824 (1995). doi:10.1016/0022-5096(95)00043-I

    Article  MathSciNet  MATH  Google Scholar 

  22. Gustafsson, B.: The convergence rate for difference approximations to mixed initial boundary value problems. Math. Comput. 29(130), 396–406 (1975)

    Article  MathSciNet  MATH  Google Scholar 

  23. Gustafsson, B., Kreiss, H.-O., Oliger, J.: Time Dependent Problems and Difference Methods. Wiley-Interscience, New York (1996)

    Google Scholar 

  24. Hagstrom, T., Mar-Or, A., Givoli, D.: High-order local absorbing conditions for the wave equation: extensions and improvements. J. Comput. Phys. 227(6), 3322–3357 (2008). doi:10.1016/j.jcp.2007.11.040

    Article  MathSciNet  MATH  Google Scholar 

  25. Kame, N., Yamashita, T.: Simulation of the spontaneous growth of a dynamic crack without constraints on the crack tip path. Geophys. J. Int. 139(2), 345–358 (1999). doi:10.1046/j.1365-246x.1999.00940.x

    Article  Google Scholar 

  26. 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, B09317 (2008). doi:10.1029/2007JB005553

    Article  Google Scholar 

  27. Knupp, P.M., Steinberg, S.: The Fundamentals of Grid Generation. CRC Press, Boca Raton (1993)

    Google Scholar 

  28. Kozdon, J.E., Dunham, E.M.: Rupture to the trench in dynamic models of the Tohoku-Oki earthquake. Abstract U51B-0041 presented at 2011 Fall Meeting, AGU, San Francisco, Calif., 5–9 Dec., 2011

  29. Kozdon, J.E., Dunham, E.M.: Rupture to the trench: dynamic rupture simulations of the 11 March 2011 Tohoku earthquake. Bull. Seism. Soc. Am. (2013, accepted). URL www.stanford.edu/~jkozdon/publications/kozdon_dunham_tohoku_BSSA12.pdf

  30. 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(2), 341–367 (2012). doi:10.1007/s10915-011-9485-3

    Article  MathSciNet  MATH  Google Scholar 

  31. Kreiss, H.-O.: Initial boundary value problems for hyperbolic systems. Commun. Pure Appl. Math. 23(3), 277–298 (1970). doi:10.1002/cpa.3160230304

    Article  MathSciNet  Google Scholar 

  32. Kreiss, H.-O., Lorenz, J.: Intial-Boundary Value Problems and the Navier-Stokes Equations. Academic Press, New York (1989)

    Google Scholar 

  33. Kreiss, H.-O., Scherer, G.: Finite element and finite difference methods for hyperbolic partial differential equations. In: Mathematical Aspects of Finite Elements in Partial Differential Equations; Proceedings of the Symposium, Madison, WI, pp. 195–212 (1974)

    Google Scholar 

  34. Kreiss, H.-O., Scherer, G.: On the existence of energy estimates for difference approximations for hyperbolic systems. Technical report, Dept. of Scientific Computing, Uppsala University (1977)

  35. Lapusta, N., Rice, J.R., Ben-Zion, Y., Zheng, G.: Elastodynamic analysis for slow tectonic loading with spontaneous rupture episodes on faults with rate- and state-dependent friction. J. Geophys. Res. 105, 23765–23790 (2000). doi:10.1029/2000JB900250

    Article  Google Scholar 

  36. Ma, S., Liu, P.: Modeling of the perfectly matched layer absorbing boundaries and intrinsic attenuation in explicit finite-element methods. Bull. Seismol. Soc. Am. 96(5), 1779–1794 (2006). doi:10.1785/0120050219

    Article  Google Scholar 

  37. Madariaga, R., Olsen, K., Archuleta, R.: Modeling dynamic rupture in a 3D earthquake fault model. Bull. Seismol. Soc. Am. 88(5), 1182–1197 (1998)

    Google Scholar 

  38. Malvern, L.E.: Introduction to the Mechanics of a Continuous Medium, 1st edn. Prentice Hall, New York (1977)

    Google Scholar 

  39. Mattsson, K., Nordström, J.: Summation by parts operators for finite difference approximations of second derivatives. J. Comput. Phys. 199(2), 503–540 (2004). doi:10.1016/j.jcp.2004.03.001

    Article  MathSciNet  MATH  Google Scholar 

  40. Miura, S., Takahashi, N., Nakanishi, A., Ito, A., Kodaira, S., Tsuru, T., Kaneda, Y.: Seismic velocity structure off Miyagi fore-arc region, Japan Trench, using ocean bottom seismographic data. In: Frontier Res. Earth Evolut., vol. 1, pp. 337–340 (2001)

    Google Scholar 

  41. Miura, S., Takahashi, N., Nakanishi, A., Tsuru, T., Kodaira, S., Kaneda, Y.: Structural characteristics off Miyagi forearc region, the Japan Trench seismogenic zone, deduced from a wide-angle reflection and refraction study. Tectonophysics 407(3–4), 165–188 (2005). doi:10.1016/j.tecto.2005.08.001

    Article  Google Scholar 

  42. Miyatake, T.: Numerical simulations of earthquake source process by a three-dimensional crack model. Part I. Rupture process. J. Phys. Earth 28(6), 565–598 (1980)

    Article  Google Scholar 

  43. Moczo, P., Kristek, J., Gallis, M., Pazak, P., Balazovjecha, M.: The finite-difference and finite-element modeling of seismic wave propagation and earthquake motion. Acta Phys. Slovaca 57(2), 177–406 (2007)

    Article  Google Scholar 

  44. Nilsson, S., Petersson, N.A., Sjogreen, B., Kreiss, H.-O.: Stable difference approximations for the elastic wave equation in second order formulation. SIAM J. Numer. Anal. 45(5), 1902–1936 (2007). doi:10.1137/060663520

    Article  MathSciNet  MATH  Google Scholar 

  45. Noda, H., Dunham, E.M., Rice, J.R.: Earthquake ruptures with thermal weakening and the operation of major faults at low overall stress levels. J. Geophys. Res. 114, B07302 (2009). doi:10.1029/2008JB006143

    Article  Google Scholar 

  46. Nordström, J.: Conservative finite difference formulations, variable coefficients, energy estimates and artificial dissipation. J. Sci. Comput. 29(3), 375–404 (2006). doi:10.1007/s10915-005-9013-4

    Article  MathSciNet  MATH  Google Scholar 

  47. Nordström, J., Carpenter, M.H.: High-order finite difference methods, multidimensional linear problems, and curvilinear coordinates. J. Comput. Phys. 173(1), 149–174 (2001). doi:10.1006/jcph.2001.6864

    Article  MathSciNet  MATH  Google Scholar 

  48. Oglesby, D.D., Archuleta, R.J., Nielsen, S.B.: Earthquakes on dipping faults: the effects of broken symmetry. Science 280(5366), 1055–1059 (1998). doi:10.1126/science.280.5366.1055

    Article  Google Scholar 

  49. Olsson, P.: Summation by parts, projections, and stability. II. Math. Comput. 64(212), 1473–1493 (1995)

    Article  MathSciNet  MATH  Google Scholar 

  50. Pelties, C., de la Puente, J., Ampuero, J.-P., Brietzke, G.B., Käser, M.: Three-dimensional dynamic rupture simulation with a high-order discontinuous Galerkin method on unstructured tetrahedral meshes. J. Geophys. Res. Solid Earth 117, B02309 (2012). doi:10.1029/2011JB008857

    Article  Google Scholar 

  51. Perrin, G., Rice, J.R., Zheng, G.: Self-healing slip pulse on a frictional surface. J. Mech. Phys. Solids 43(9), 1461–1495 (1995). doi:10.1016/0022-5096(95)00036-I

    Article  MathSciNet  MATH  Google Scholar 

  52. Rice, J.R.: Constitutive relations for fault slip and earthquake instabilities. Pure Appl. Geophys. 121(3), 443–475 (1983). doi:10.1007/BF02590151

    Article  Google Scholar 

  53. Rice, J.R., Ruina, A.L.: Stability of steady frictional slipping. J. Appl. Mech. 50(2), 343–349 (1983). doi:10.1115/1.3167042

    Article  MATH  Google Scholar 

  54. Rice, J.R., Lapusta, N., Ranjith, K.: Rate and state dependent friction and the stability of sliding between elastically deformable solids. J. Mech. Phys. Solids 49(9), 1865–1898 (2001). doi:10.1016/S0022-5096(01)00042-4

    Article  MATH  Google Scholar 

  55. Roache, P.J.: Verification and Validation in Computational Science and Engineering. Hermosa Publishers, Albuquerque (1998)

    Google Scholar 

  56. Rojas, O., Dunham, E.M., Day, S.M., Dalguer, L.A., Castillo, J.E.: Finite difference modelling of rupture propagation with strong velocity-weakening friction. Geophys. J. Int. 179, 1831–1858 (2009). doi:10.1111/j.1365-246X.2009.04387.x

    Article  Google Scholar 

  57. Sato, M., Ishikawa, T., Ujihara, N., Yoshida, S., Fujita, M., Mochizuki, M., Asada, A.: Displacement above the hypocenter of the 2011 Tohoku-Oki earthquake. Science 332(6036) (2011). doi:10.1126/science.1207401

  58. Slaughter, W.S.: The Linearized Theory of Elasticity. Birkhäuser, Boston (2002)

    Book  MATH  Google Scholar 

  59. Strand, B.: Summation by parts for finite difference approximations for d/dx. J. Comput. Phys. 110(1), 47–67 (1994). doi:10.1006/jcph.1994.1005

    Article  MathSciNet  MATH  Google Scholar 

  60. Svärd, M., Nordström, J.: On the order of accuracy for difference approximations of initial-boundary value problems. J. Comput. Phys. 218(1), 333–352 (2007). doi:10.1016/j.jcp.2006.02.014

    Article  Google Scholar 

  61. Tessmer, E., Kosloff, D., Behle, A.: Elastic wave propagation simulation in the presence of surface topography. Geophys. J. Int. 108(2), 621–632 (1992). doi:10.1111/j.1365-246X.1992.tb04641.x

    Article  Google Scholar 

  62. Zhang, H., Chen, X.: Dynamic rupture process of the 1999 Chi-Chi, Taiwan, earthquake. Earth Sci. 22(1), 3–12 (2009). doi:10.1007/s11589-009-0003-8

    Article  Google Scholar 

Download references

Acknowledgements

J.E.K. and E.M.D. were supported by National Science Foundation (NSF) award EAR-0910574 and the Southern California Earthquake Center (SCEC), as funded by NSF Cooperative Agreement EAR-0529922 and US Geological Survey Cooperative Agreement 07HQAG0008 (SCEC contribution number 1424). J.E.K. was also supported by NSF Fellowship for Transformative Computational Science using CyberInfrastructure OCI-1122734. The computations in this paper were conducted at the Stanford Center for Computational Earth and Environmental Science. Finally, we would like to thank the two anonymous reviewers of the manuscript for their suggestions and comments.

Author information

Authors and Affiliations

  1. Department of Geophysics, Stanford University, Stanford, CA, USA

    Jeremy E. Kozdon

  2. Department of Geophysics and Institute for Computational and Mathematical Engineering, Stanford University, Stanford, CA, USA

    Eric M. Dunham

  3. Department of Mathematics, Linköping University, Linköping, Sweden

    Jan Nordström

Authors
  1. Jeremy E. Kozdon
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Eric M. Dunham
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Jan Nordström
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Jeremy E. Kozdon.

Appendix: Equivalence of Friction Law in Physical and Characteristic Variables

Appendix: Equivalence of Friction Law in Physical and Characteristic Variables

In this appendix we show that if ∂f/∂V≥0 then friction law f(V,ψ) is expressible in characteristic form. Assuming that the interface is between blocks (a) and (b), the normal stress and fault normal velocity can be written using the characteristic interface variables (30):

(114)
(115)

Continuity of normal stress (36) and fault normal velocity (38) implies that these can be written as

(116)

All that remains is to show that (40) can also be written in characteristic form (35). We can write the shear stress and slip velocity using the characteristic variables (31):

(117)
(118)
(119)
(120)

Force balance (36) and the nonlinear friction law (40) then can be written as the nonlinear system

(121)

where \(V_{m} = -v_{m}^{(a)} - v_{m}^{(b)}\) and \(V_{z} = -v_{z}^{(a)} + v_{z}^{(b)}\); see (39). The Jacobian of (121) with respect to the variables \(\mathcal{W}^{-(a)}_{m}\), \(\mathcal{W}^{-(a)}_{z}\), \(\mathcal{W}^{-(b)}_{m}\), and \(\mathcal{W}^{-(b)}_{z}\), is

(122)

where 0 2 is the 2×2 zero matrix and

(123)
(124)

with f=f(V,ψ). The determinant of J is

(125)

If ∂f/∂V≥0 then J≠0 and it follows by the implicit function theorem that friction law f is expressible in characteristic form.

Rights and permissions

Reprints and permissions

About this article

Cite this article

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, 92–124 (2013). https://doi.org/10.1007/s10915-012-9624-5

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10915-012-9624-5

Keywords

Search

Navigation