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Tsunami modelling with adaptively refined finite volume methods*

Published online by Cambridge University Press:  28 April 2011

Randall J. LeVeque ,
David L. George  and
Marsha J. Berger
Randall J. LeVeque
Affiliation:
Department of Applied Mathematics, University of Washington, Seattle, WA 98195-2420, USA E-mail: rjl@uw.edu
David L. George
Affiliation:
US Geological Survey, Cascades Volcano Observatory, Vancouver, WA 98683, USA E-mail: dgeorge@usgs.gov
Marsha J. Berger
Affiliation:
Courant Institute of Mathematical Sciences, New York University, NY 10012, USA E-mail: berger@cims.nyu.edu
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Abstract

Numerical modelling of transoceanic tsunami propagation, together with the detailed modelling of inundation of small-scale coastal regions, poses a number of algorithmic challenges. The depth-averaged shallow water equations can be used to reduce this to a time-dependent problem in two space dimensions, but even so it is crucial to use adaptive mesh refinement in order to efficiently handle the vast differences in spatial scales. This must be done in a ‘wellbalanced’ manner that accurately captures very small perturbations to the steady state of the ocean at rest. Inundation can be modelled by allowing cells to dynamically change from dry to wet, but this must also be done carefully near refinement boundaries. We discuss these issues in the context of Riemann-solver-based finite volume methods for tsunami modelling. Several examples are presented using the GeoClaw software, and sample codes are available to accompany the paper. The techniques discussed also apply to a variety of other geophysical flows.

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Type
Research Article
Information
Acta Numerica , Volume 20 , May 2011 , pp. 211 - 289
Copyright
Copyright © Cambridge University Press 2011

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References

REFERENCES

Atwater, B. F.et al. (2005), The Orphan Tsunami of 1700, University of Washington Press, Seattle.Google Scholar
Bale, D. S., LeVeque, R. J., Mitran, S. and Rossmanith, J. A. (2002), ‘A wave propagation method for conservation laws and balance laws with spatially varying flux functions’, SIAM J. Sci. Comput. 24, 955–978.CrossRefGoogle Scholar
Bardet, J. P., Synolakis, C. E., Davies, H. L., Imamura, F. and Okal, E. A. (2003), ‘Landslide tsunamis: Recent findings and research directions’, Pure Appl. Geophys. 160, 1793–1809.CrossRefGoogle Scholar
Berger, M. and Oliger, J. (1984), ‘Adaptive mesh refinement for hyperbolic partial differential equations’, J.Comput. Phys. 53, 484–512.CrossRefGoogle Scholar
Berger, M. J. and Colella, P. (1989), ‘Local adaptive mesh refinement for shock hydrodynamics’, J. Comput. Phys. 82, 64–84.CrossRefGoogle Scholar
Berger, M. J. and LeVeque, R. J. (1998), ‘Adaptive mesh refinement using wave-propagation algorithms for hyperbolic systems’, SIAM J. Numer. Anal. 35, 2298–2316.CrossRefGoogle Scholar
Berger, M. J. and Rigoutsos, I. (1991), ‘An algorithm for point clustering and grid generation’, IEEE Trans. Sys. Man & Cyber. 21, 1278–1286.CrossRefGoogle Scholar
Berger, M. J., Calhoun, D. A., Helzel, C. and LeVeque, R. J. (2009), ‘Logically rectangular finite volume methods with adaptive refinement on the sphere’, Phil. Trans. R. Soc. A 367, 4483–4496.CrossRefGoogle Scholar
Berger, M. J., George, D. L., LeVeque, R. J. and Mandli, K. T. (2010), The GeoClaw software for depth-averaged flows with adaptive refinement. To appear in Advances in Water Resources. Available at: www.clawpack.org/links/awr11.Google Scholar
Bona, J. L., Chen, M. and Saut, J.-C. (2002), ‘Boussinesq equations and other systems for small-amplitude long waves in nonlinear dispersive media I: Derivation and linear theory’, J. Nonlinear Sci. 12, 283–318.CrossRefGoogle Scholar
Botta, N., Klein, R., Langenberg, S. and Lützenkirchen, S. (2004), ‘Well balanced finite volume methods for nearly hydrostatic flows’, J. Comput. Phys. 196, 539–565.CrossRefGoogle Scholar
Bouchut, F (2004), Nonlinear Stability of Finite Volume Methods for Hyperbolic Conservation Laws and Well-Balanced Schemes for Sources, Birkhäuser.CrossRefGoogle Scholar
Bourgeois, J (2009), Geologic effects and records of tsunamis. In The Sea, Vol. 15 (Bernard, E. N and Robinson, A. R., eds), Harvard University Press, pp. 55–92.Google Scholar
Burwell, D., Tolkova, E. and Chawla, A. (2007), ‘Diffusion and dispersion characterization of a numerical tsunami model’, Ocean Modelling 19, 10–30.CrossRefGoogle Scholar
Carrier, G. F. and Yeh, H. (2005), ‘Tsunami propagation from a finite source’, CMES 10, 113–121.Google Scholar
Carrier, G. F., Wu, T. T. and Yeh, H. (2003), ‘Tsunami run-up and draw-down on a plane beach’, J. Fluid Mech. 475, 79–99.CrossRefGoogle Scholar
Castro, M. J., LeFloch, P. G., Munoz, M. L. and Par, C.és (2008), ‘Why many theories of shock waves are necessary: Convergence error in formally path-consistent schemes’, J. Comput. Phys. 227, 8107–8129.CrossRefGoogle Scholar
Costa, A. and Macedonio, G. (2005), ‘Numerical simulation of lava flows based on depth-averaged equations’, Geophys. Res. Lett. 32, L05304.CrossRefGoogle Scholar
Dawson, A., Long, D. and Smith, D. (1988), ‘The Storegga Slides: Evidence from eastern Scotland for a possible tsunami’, Marine Geology, January 1988.CrossRefGoogle Scholar
Denlinger, R. P. and Iverson, R. M. (2004 a), ‘Granular avalanches across irregular three-dimensional terrain 1: Theory and computation’, J. Geophys. Res. 109, F01014.Google Scholar
Denlinger, R. P. and Iverson, R. M. (2004 b), ‘Granular avalanches across irregular three-dimensional terrain 2: Experimental tests’, J. Geophys. Res. 109, F01015.Google Scholar
Einfeldt, B. (1988), ‘On Godunov-type methods for gas dynamics’, SIAM J. Numer. Anal. 25, 294–318.CrossRefGoogle Scholar
Einfeldt, B., Munz, C. D., Roe, P. L. and Sjogreen, B. (1991), ‘On Godunov type methods near low densities’, J. Comput. Phys. 92, 273–295.CrossRefGoogle Scholar
Fomel, S. and Claerbout, J. F. (2009), ‘Guest editors’ introduction: Reproducible research’, Comput. Sci. Engrg 11, 5–7.CrossRefGoogle Scholar
Gallardo, J. M., Parés, C. and Castro, M. (2007), ‘On a well-balanced high-order finite volume scheme for shallow water equations with topography and dry areas’, J. Comput. Phys. 227, 574–601.CrossRefGoogle Scholar
Geist, E. L. and Parsons, T. (2006), ‘Probabilistic analysis of tsunami hazards’, Nat. Haz. 37, 277–314.CrossRefGoogle Scholar
Geist, E. L., Parsons, T., ten Brink, U. S and Lee, H. J. (2009), ‘Tsunami probability’, 15, 201–235.Google Scholar
Gelfenbaum, G. and Jaffe, B. (2003), ‘Erosion and sedimentation from the 17 July, 1998 Papua New Guinea tsunami’, Pure Appl. Geophys. 160, 1969–1999.CrossRefGoogle Scholar
George, D. (2010), ‘Adaptive finite volume methods with well-balanced Riemann solvers for modeling floods in rugged terrain: Application to the Malpasset dam-break flood (France, 1959)’, Int. J. Numer. Meth. Fluids.Google Scholar
George, D. L. (2006), Finite volume methods and adaptive refinement for tsunami propagation and inundation. PhD thesis, University of Washington.Google Scholar
George, D. L. (2008), ‘Augmented Riemann solvers for the shallow water equations over variable topography with steady states and inundation’, J. Comput. Phys. 227, 3089–3113.CrossRefGoogle Scholar
George, D. L. and LeVeque, R. J. (2006), ‘Finite volume methods and adaptive refinement for global tsunami propagation and local inundation’, Science of Tsunami Hazards 24, 319–328.Google Scholar
González, F. I. and Kulikov, Y. A. (1993), Tsunami dispersion observed in the deep ocean. In Tsunamis in the World (Tinti, S., ed.), Vol. 1 of Advances in Natural and Technological Hazards Research, Kluwer, pp. 7–16.CrossRefGoogle Scholar
González, F. I., Geist, E. L., Jaffe, B., Kanoglu, U.et al. (2009), ‘Probabilistic tsunami hazard assessment at Seaside, Oregon, for near- and far-field seismic sources’, J. Geophys. Res. 114, C11023.Google Scholar
Gosse, L. (2000), ‘A well-balanced flux-vector splitting scheme designed for hyperbolic systems of conservation laws with source terms’, Comput. Math. Appl. 39, 135–159.CrossRefGoogle Scholar
Gosse, L. (2001), ‘A well-balanced scheme using non-conservative products designed for hyperbolic systems of conservation laws with source terms’, Math. Mod. Meth. Appl. Sci. 11, 339–365.CrossRefGoogle Scholar
Greenberg, J. M. and LeRoux, A. Y. (1996), ‘A well-balanced scheme for numerical processing of source terms in hyperbolic equations’, SIAM J. Numer. Anal. 33, 1–16.CrossRefGoogle Scholar
Grilli, S. T., Ioualalen, M., Asavanant, J., Shi, F., Kirby, J. T. and Watts, P. (2007), ‘Source constraints and model simulation of the December 26, 2004, Indian Ocean Tsunami’, J. Waterway, Port, Coastal, and Ocean Engineering 133, 414.CrossRefGoogle Scholar
Haflidason, H., Sejrup, H., Nygård, A., Mienert, J. and Bryn, P. (2004), ‘The Storegga Slide: Architecture, geometry and slide development’, Marine Geology, January 2004.CrossRefGoogle Scholar
Hammack, J. and Segur, H. (1978), ‘Modelling criteria for long water waves’, J. Fluid Mech. 84, 337–358.CrossRefGoogle Scholar
Harten, A., Lax, P. D. and van Leer, B. (1983), ‘On upstream differencing and Godunov-type schemes for hyperbolic conservation laws’, SIAM Review 25, 35–61.CrossRefGoogle Scholar
Higman, B., Gelfenbaum, G., Lynett, P., Moore, A. and Jaffe, B. (2007), Predicted sedimentary record of reflected bores. In Proc. Sixth International Symposium on Coastal Engineering and Science of Coastal Sediment Processes, ASCE, pp. 1–14.Google Scholar
Hirata, K., Geist, E., Satake, K., Tanioka, Y. and Yamaki, S. (2003), ‘Slip distribution of the 1952 Tokachi-Oki earthquake (M 8.1) along the Kuril Trench deduced from tsunami waveform inversion’, J.Geophys. Res.CrossRefGoogle Scholar
Huntington, K., Bourgeois, J., Gelfenbaum, G., Lynett, P., Jaffe, B., Yeh, H. and Weiss, R. (2007), ‘Sandy signs of a tsunami's onshore depth and speed’, EOS 88, 577–578. www.agu.org/journals/eo/eo0752/2007EO52-tabloid.pdf.CrossRefGoogle Scholar
In, A. (1999), ‘Numerical evaluation of an energy relaxation method for inviscid real fluids’, SIAM J. Sci. Comput. 21, 340–365.CrossRefGoogle Scholar
Jankaew, K., Atwater, B. F., Sawai, Y., Choowong, M., Charoentitirat, T., Martin, M. E and Prendergast, A. (2008), ‘Medieval forewarning of the 2004 Indian Ocean tsunami in Thailand’, Nature 455, 1228–1231.CrossRefGoogle Scholar
Kelsey, H. M., Nelson, A. R., Hemphill-Haley, E. and Witter, R. C. (2005), ‘Tsunami history of an Oregon coastal lake reveals a 4600 yr record of great earthquakes on the Cascadia subduction zone’, GSA Bulletin 117, 1009–1032.CrossRefGoogle Scholar
Kowalik, Z., Knight, W., Logan, T. and Whitmore, P. (2005), ‘Modeling of the global tsunami: Indonesian Tsunami of 26 December 2004.’, Science of Tsunami Hazards 23, 40–56.Google Scholar
Langseth, J. O. and LeVeque, R. J. (2000), ‘A wave-propagation method for three-dimensional hyperbolic conservation laws’, J.Comput. Phys. 165, 126–166.CrossRefGoogle Scholar
LeVeque, R. J. (1996), ‘High-resolution conservative algorithms for advection in incompressible flow’, SIAM J. Numer. Anal. 33, 627–665.CrossRefGoogle Scholar
LeVeque, R. J. (2002), Finite Volume Methods for Hyperbolic Problems, Cambridge University Press.CrossRefGoogle Scholar
LeVeque, R. J. (2009), ‘Python tools for reproducible research on hyperbolic problems’, Comput. Sci. Engrg 11, 19–27.CrossRefGoogle Scholar
LeVeque, R. J. (2010), ‘A well-balanced path-integral f-wave method for hyperbolic problems with source terms’, J.Sci. Comput. doi:10.1007/s10915–010–9411–0. www.clawpack.org/links/wbfwave10.Google Scholar
LeVeque, R. J. and George, D. L. (2004), High-resolution finite volume methods for the shallow water equations with bathymetry and dry states. In Proc. Long-Wave Workshop, Catalina (Liu, P. L.-F., Yeh, H. and Synolakis, C., eds), Vol. 10, World Scientific, pp. 43–73. www.amath.washington.edu/~rjl/pubs/catalina04/.Google Scholar
LeVeque, R. J. and Pelanti, M. (2001), ‘A class of approximate Riemann solvers and their relation to relaxation schemes’, J.Comput. Phys. 172, 572–591.CrossRefGoogle Scholar
Liu, P. L., Yeh, H. and Synolakis, C., eds (2008), Advanced Numerical Models for Simulating Tsunami Waves and Runup, Vol. 10 of Advances in Coastal and Ocean Engineering, World Scientific.Google Scholar
Liu, P., Lynett, P., Fernando, H., Jaffe, B. and Fritz, H. (2005), ‘Observations by the International Tsunami Survey Team in Sri Lanka’, Science 308, 1595.CrossRefGoogle ScholarPubMed
Liu, P., Woo, S. and Cho, Y. (1998), ‘Computer programs for tsunami propagation and inundation’. ceeserver.cee.cornell.edu/pll-group/comcot.htm.Google Scholar
Lynett, P. and Liu, P. L. (2002), ‘A numerical study of submarine-landslide-generated waves and run-up’, Proc. Royal Soc. London Ser. A 458, 2885–2910.CrossRefGoogle Scholar
Mader, C. L. and Gittings, M. L. (2002), ‘Modeling the 1958 Lituya Bay mega tsunami, II’, Science of Tsunami Hazards 20, 241.Google Scholar
Mandli, K. T. (2010), Personal communication.Google Scholar
Mansinha, L. and Smylie, D. (1971), ‘The displacement fields of inclined faults’, Bull. Seism. Soc. Amer. 61, 1433–1438.CrossRefGoogle Scholar
Martin, M. E., Weiss, R., Bourgeois, J., Pinegina, T. K., Houston, H. and Titov, V. V. (2008), ‘Combining constraints from tsunami modeling and sedimentology to untangle the 1969 Ozernoi and 1971 Kamchatskii tsunamis’, Geophys. Res. Lett. 35, L01610.CrossRefGoogle Scholar
Masson, D. G., Harbitz, C. B., Wynn, R. B., Pedersen, G. and Løvholt, F. (2006), ‘Submarine landslides: processes, triggers and hazard prediction’, Philos. Trans. Royal Soc. A: Math. Phys. Engrg Sci. 364, 2009.CrossRefGoogle ScholarPubMed
McCulloch, D. S. (1966), Slide-induced waves, seiching and ground fracturing caused by the earthquake of March 27, 1964, at Kenai Lake, Alaska. USGS Professional Paper 543–A. www.dggs.dnr.state.ak.us/pubs/pubs?reqtype=citation&ID=3884.CrossRefGoogle Scholar
Meinig, C., Stalin, S. E., Nakamura, A. I., Gonźlez, F. and Milburn, H. B. (2006), Technology developments in real-time tsunami measuring, monitoring and forecasting. In OCEANS, 2005: Proc. MTS/IEEE, pp. 1673–1679.Google Scholar
Merali, Z. 2010), ‘Why scientific computing does not compute’, Nature 467, 775–777.CrossRefGoogle Scholar
Miller, D. J. (1960), Giant waves in Lituya Bay, Alaska. USGS Professional Paper 354-C. www.dggs.dnr.state.ak.us/pubs/pubs?reqtype=citation&ID=3852.CrossRefGoogle Scholar
Noelle, S., Pankrantz, N., Puppo, G. and Natvig, J. R. (2006), ‘Well-balanced finite volume schemes of arbitrary order of accuracy for shallow water flows’, J.Comput. Phys. 213, 474–499.CrossRefGoogle Scholar
Okada, Y. (1985), ‘Surface deformation due to shear and tensile faults in a halfspace’, Bull. Seism. Soc. Amer. 75, 1135–1154.CrossRefGoogle Scholar
Okada, Y. (1992), ‘Internal deformation due to shear and tensile faults in a halfspace’, Bull. Seism. Soc. Amer. 82, 1018–1040.CrossRefGoogle Scholar
Ostapenko, V. V. (1999), ‘Numerical simulation of wave flows caused by a shoreside landslide’, J. Applied Mech. Tech. Phys. 40, 647–654.CrossRefGoogle Scholar
Pelanti, M., Bouchut, F. and Mangeney, A. (2008), ‘A Roe-type scheme for two-phase shallow granular flows over variable topography’, M2AN 42, 851–885.CrossRefGoogle Scholar
Pelanti, M., Bouchut, F. and Mangeney, A. (2011), ‘A Riemann solver for singlephase and two-phase shallow flow models based on relaxation: Relations with Roe and VFRoe solvers’, J. Comput. Phys. 230, 515–550.CrossRefGoogle Scholar
Percival, D. B., Denbo, D. W., Eblé, M. C., Gica, E., Mofjeld, H. O., Spillane, M. C., Tang, L. and Titov, V. V. (2010), ‘Extraction of tsunami source coefficients via inversion of DART buoy data’, Natural Hazards doi:10.1007/s11069–010–9688–1.Google Scholar
Plafker, G., Kachadoorian, R., Eckel, E. B. and Mayo, L. R. (1969), Effects of the earthquake of March 27, 1964 on various communities. USGS Professional Paper 542G. www.dggs.dnr.state.ak.us/pubs/pubs?reqtype=citation&ID=3883.Google Scholar
Quirk, J. J. (2003), Computational science: ‘Same old silence, same old mistakes’ ‘Something more is needed…’. In Adaptive Mesh Refinement: Theory and Applications (Plewa, T., Linde, T. and Weirs, V. G., eds), Vol. 41 of Lecture Notes in Computational Science and Engineering, Springer, pp. 3–28.CrossRefGoogle Scholar
Roache, P. J. (1998), Verification and Validation in Computational Science and Engineering, Hermosa Publishers, Albuquerque, NM.Google Scholar
Saito, T., Matsuzawa, T., Obara, K. and Baba, T. (2010), ‘Dispersive tsunami of the 2010 Chile earthquake recorded by the high-sampling-rate ocean-bottom pressure gauges’, Geophys. Res. Lett. 37, L22303.CrossRefGoogle Scholar
Satake, K., Shimazaki, K., Tsuji, Y. and Ueda, K. (1996), ‘Time and size of a giant earthquake in Cascadia inferred from Japanese tsunami records of January 1700’, Nature 379, 246–249.CrossRefGoogle Scholar
Satake, K., Wang, K. and Atwater, B. (2003), ‘Fault slip and seismic moment of the 1700 Cascadia earthquake inferred from Japanese tsunami descriptions’, J. Geophys. Res. 108, 2535.Google Scholar
Synolakis, C., Bardet, J., Borrero, J., Davies, H., Okal, E., Silver, E., Sweet, S. and Tappin, D. (2002), ‘The slump origin of the 1998 Papua New Guinea tsunami’, Proc. Royal Soc. London Ser. A: Math. Phys. Engrg Sci. 458, 763.CrossRefGoogle Scholar
Synolakis, C. E. and Bernard, E. N. (2006), ‘Tsunami science before and beyond Boxing Day 2004’, Philos. Trans. Royal Soc. A: Math. Phys. Engrg Sci. 364, 2231.CrossRefGoogle ScholarPubMed
Synolakis, C. E., Bernard, E. N., Titov, V. V., Kânoğlu, U. and González, F. I. (2008), ‘Validation and verification of tsunami numerical models’, Pure Appl. Geophys. 165, 2197–2228.CrossRefGoogle Scholar
Thacker, W. C. (1981), ‘Some exact solutions to the nonlinear shallow water wave equations’, J. Fluid Mech. 107, 499–508.CrossRefGoogle Scholar
Titov, V. V. and Synolakis, C. E. (1995), ‘Modeling of breaking and nonbreaking long wave evolution and runup using VTCS-2’, J. Waterways, Ports, Coastal and Ocean Engineering 121, 308–316.CrossRefGoogle Scholar
Titov, V. V. and Synolakis, C. E. (1998), ‘Numerical modeling of tidal wave runup’, J. Waterways, Ports, Coastal and Ocean Engineering 124, 157–171.CrossRefGoogle Scholar
Titov, V. V., Gonzalez, F. I., Bernard, E. N., Eble, M. C., Mofjeld, H. O., Newman, J. C. and Venturato, A. J. (2005), ‘Real-time tsunami forecasting: Challenges and solutions’, Nat. Hazards 35, 35–41.CrossRefGoogle Scholar
Toro, E. F. (2001), Shock-Capturing Methods for Free-Surface Shallow Flows, Wiley.Google Scholar
Wang, X. and Liu, P. L. (2007), ‘Numerical simulations of the 2004 Indian Ocean tsunamis: Coastal effects’, J. Earthquake Tsunami 1, 273–297.CrossRefGoogle Scholar
Watts, P., Grilli, S., Kirby, J., Fryer, G. J. and Tappin, D. R. (2003), ‘Landslide tsunami case studies using a Boussinesq model and a fully nonlinear tsunami generation model’, Nat. Haz. Earth Sys. Sci. 3, 391–402.CrossRefGoogle Scholar
Weiss, R., Fritz, H. and Wünnemann, K. (2009), ‘Hybrid modeling of the megatsunami runup in Lituya Bay after half a century’, Geophys. Res. Lett. 36, L09602.CrossRefGoogle Scholar
Yeh, H., Chadha, R. K., Francis, M., Katada, T., Latha, G., Peterson, C., Raghu-ramani, G. and Singh, J. P. (2006), ‘Tsunami runup survey along the southeast Indian coast’, Earthquake Spectra 22, S173–S186.CrossRefGoogle Scholar
Yeh, H., Liu, P. L. and Synolakis, C., eds (1996), Long-Wave Runup Models, World Scientific.Google Scholar
Yeh, H., Liu, P., Briggs, M. and Synolakis, C. (1994), ‘Propagation and amplification of tsunamis at coastal boundaries’, Nature 372, 353–355.CrossRefGoogle Scholar

Online references

www1: AMROC software: amroc.sourceforge.net/.Google Scholar
www2: Chombo software: seesar.lbl.gov/anag/chombo/.Google Scholar
www3: Clawpack software: www.clawpack.org.Google Scholar
www4: COMCOT software: ceeserver.cee.cornell.edu/pll-group/comcot.htm.Google Scholar
www5: DART data: www.ndbc.noaa.gov/.Google Scholar
www6: FLASH software: flash.uchicago.edu/website/home/.Google Scholar
www7: GeoClaw software: www.clawpack.org/geoclaw.Google Scholar
www8: Hilo, HI 1/3 arc-second MHW Tsunami Inundation DEM: www.ngdc.noaa.gov/mgg/inundation/.Google Scholar
www9: National Geophysical Data Center (NGDC) GEODAS: www.ngdc.noaa.gov/mgg/gdas/gd_designagrid.html.Google Scholar
www10: NOAA Tsunami Inundation Digital Elevation Models (DEMs): www.ngdc.noaa.gov/mgg/inundation/tsunami/.Google Scholar
http://www11: SAMRAI: computation.llnl.gov/casc/SAMRAI/.Google Scholar
www12: USGS source for 27 February 2010 earthquake: earthquake.usgs.gov/earthquakes/eqinthenews/2010/us2010tfan/.Google Scholar
www13: Webpage for this paper: www.clawpack.org/links/an11.Google Scholar