Skip to main content Accessibility help

Tsunami modelling with adaptively refined finite volume methods*

  • Randall J. LeVeque (a1), David L. George (a2) and Marsha J. Berger (a3)

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.

Hide All
Atwater, B. al. (2005), The Orphan Tsunami of 1700, University of Washington Press, Seattle.
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, 955978.
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, 17931809.
Berger, M. and Oliger, J. (1984), ‘Adaptive mesh refinement for hyperbolic partial differential equations’, J.Comput. Phys. 53, 484512.
Berger, M. J. and Colella, P. (1989), ‘Local adaptive mesh refinement for shock hydrodynamics’, J. Comput. Phys. 82, 6484.
Berger, M. J. and LeVeque, R. J. (1998), ‘Adaptive mesh refinement using wave-propagation algorithms for hyperbolic systems’, SIAM J. Numer. Anal. 35, 22982316.
Berger, M. J. and Rigoutsos, I. (1991), ‘An algorithm for point clustering and grid generation’, IEEE Trans. Sys. Man & Cyber. 21, 12781286.
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, 44834496.
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:
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, 283318.
Botta, N., Klein, R., Langenberg, S. and Lützenkirchen, S. (2004), ‘Well balanced finite volume methods for nearly hydrostatic flows’, J. Comput. Phys. 196, 539565.
Bouchut, F (2004), Nonlinear Stability of Finite Volume Methods for Hyperbolic Conservation Laws and Well-Balanced Schemes for Sources, Birkhäuser.
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. 5592.
Burwell, D., Tolkova, E. and Chawla, A. (2007), ‘Diffusion and dispersion characterization of a numerical tsunami model’, Ocean Modelling 19, 1030.
Carrier, G. F. and Yeh, H. (2005), ‘Tsunami propagation from a finite source’, CMES 10, 113121.
Carrier, G. F., Wu, T. T. and Yeh, H. (2003), ‘Tsunami run-up and draw-down on a plane beach’, J. Fluid Mech. 475, 7999.
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, 81078129.
Costa, A. and Macedonio, G. (2005), ‘Numerical simulation of lava flows based on depth-averaged equations’, Geophys. Res. Lett. 32, L05304.
Dawson, A., Long, D. and Smith, D. (1988), ‘The Storegga Slides: Evidence from eastern Scotland for a possible tsunami’, Marine Geology, January 1988.
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.
Denlinger, R. P. and Iverson, R. M. (2004 b), ‘Granular avalanches across irregular three-dimensional terrain 2: Experimental tests’, J. Geophys. Res. 109, F01015.
Einfeldt, B. (1988), ‘On Godunov-type methods for gas dynamics’, SIAM J. Numer. Anal. 25, 294318.
Einfeldt, B., Munz, C. D., Roe, P. L. and Sjogreen, B. (1991), ‘On Godunov type methods near low densities’, J. Comput. Phys. 92, 273295.
Fomel, S. and Claerbout, J. F. (2009), ‘Guest editors’ introduction: Reproducible research’, Comput. Sci. Engrg 11, 57.
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, 574601.
Geist, E. L. and Parsons, T. (2006), ‘Probabilistic analysis of tsunami hazards’, Nat. Haz. 37, 277314.
Geist, E. L., Parsons, T., ten Brink, U. S and Lee, H. J. (2009), ‘Tsunami probability’, 15, 201235.
Gelfenbaum, G. and Jaffe, B. (2003), ‘Erosion and sedimentation from the 17 July, 1998 Papua New Guinea tsunami’, Pure Appl. Geophys. 160, 19691999.
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.
George, D. L. (2006), Finite volume methods and adaptive refinement for tsunami propagation and inundation. PhD thesis, University of Washington.
George, D. L. (2008), ‘Augmented Riemann solvers for the shallow water equations over variable topography with steady states and inundation’, J. Comput. Phys. 227, 30893113.
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, 319328.
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. 716.
González, F. I., Geist, E. L., Jaffe, B., Kanoglu, al. (2009), ‘Probabilistic tsunami hazard assessment at Seaside, Oregon, for near- and far-field seismic sources’, J. Geophys. Res. 114, C11023.
Gosse, L. (2000), ‘A well-balanced flux-vector splitting scheme designed for hyperbolic systems of conservation laws with source terms’, Comput. Math. Appl. 39, 135159.
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, 339365.
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, 116.
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.
Haflidason, H., Sejrup, H., Nygård, A., Mienert, J. and Bryn, P. (2004), ‘The Storegga Slide: Architecture, geometry and slide development’, Marine Geology, January 2004.
Hammack, J. and Segur, H. (1978), ‘Modelling criteria for long water waves’, J. Fluid Mech. 84, 337358.
Harten, A., Lax, P. D. and van Leer, B. (1983), ‘On upstream differencing and Godunov-type schemes for hyperbolic conservation laws’, SIAM Review 25, 3561.
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. 114.
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.
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, 577578.
In, A. (1999), ‘Numerical evaluation of an energy relaxation method for inviscid real fluids’, SIAM J. Sci. Comput. 21, 340365.
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, 12281231.
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, 10091032.
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, 4056.
Langseth, J. O. and LeVeque, R. J. (2000), ‘A wave-propagation method for three-dimensional hyperbolic conservation laws’, J.Comput. Phys. 165, 126166.
LeVeque, R. J. (1996), ‘High-resolution conservative algorithms for advection in incompressible flow’, SIAM J. Numer. Anal. 33, 627665.
LeVeque, R. J. (2002), Finite Volume Methods for Hyperbolic Problems, Cambridge University Press.
LeVeque, R. J. (2009), ‘Python tools for reproducible research on hyperbolic problems’, Comput. Sci. Engrg 11, 1927.
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.
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. 4373.
LeVeque, R. J. and Pelanti, M. (2001), ‘A class of approximate Riemann solvers and their relation to relaxation schemes’, J.Comput. Phys. 172, 572591.
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.
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.
Liu, P., Woo, S. and Cho, Y. (1998), ‘Computer programs for tsunami propagation and inundation’.
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, 28852910.
Mader, C. L. and Gittings, M. L. (2002), ‘Modeling the 1958 Lituya Bay mega tsunami, II’, Science of Tsunami Hazards 20, 241.
Mandli, K. T. (2010), Personal communication.
Mansinha, L. and Smylie, D. (1971), ‘The displacement fields of inclined faults’, Bull. Seism. Soc. Amer. 61, 14331438.
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.
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.
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.
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. 16731679.
Merali, Z. 2010), ‘Why scientific computing does not compute’, Nature 467, 775777.
Miller, D. J. (1960), Giant waves in Lituya Bay, Alaska. USGS Professional Paper 354-C.
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, 474499.
Okada, Y. (1985), ‘Surface deformation due to shear and tensile faults in a halfspace’, Bull. Seism. Soc. Amer. 75, 11351154.
Okada, Y. (1992), ‘Internal deformation due to shear and tensile faults in a halfspace’, Bull. Seism. Soc. Amer. 82, 10181040.
Ostapenko, V. V. (1999), ‘Numerical simulation of wave flows caused by a shoreside landslide’, J. Applied Mech. Tech. Phys. 40, 647654.
Pelanti, M., Bouchut, F. and Mangeney, A. (2008), ‘A Roe-type scheme for two-phase shallow granular flows over variable topography’, M2AN 42, 851885.
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, 515550.
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.
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.
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. 328.
Roache, P. J. (1998), Verification and Validation in Computational Science and Engineering, Hermosa Publishers, Albuquerque, NM.
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.
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, 246249.
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.
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.
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.
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, 21972228.
Thacker, W. C. (1981), ‘Some exact solutions to the nonlinear shallow water wave equations’, J. Fluid Mech. 107, 499508.
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, 308316.
Titov, V. V. and Synolakis, C. E. (1998), ‘Numerical modeling of tidal wave runup’, J. Waterways, Ports, Coastal and Ocean Engineering 124, 157171.
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, 3541.
Toro, E. F. (2001), Shock-Capturing Methods for Free-Surface Shallow Flows, Wiley.
Wang, X. and Liu, P. L. (2007), ‘Numerical simulations of the 2004 Indian Ocean tsunamis: Coastal effects’, J. Earthquake Tsunami 1, 273297.
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, 391402.
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.
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.
Yeh, H., Liu, P. L. and Synolakis, C., eds (1996), Long-Wave Runup Models, World Scientific.
Yeh, H., Liu, P., Briggs, M. and Synolakis, C. (1994), ‘Propagation and amplification of tsunamis at coastal boundaries’, Nature 372, 353355.
www1: AMROC software:
www2: Chombo software:
www3: Clawpack software:
www5: DART data:
www6: FLASH software:
www7: GeoClaw software:
www8: Hilo, HI 1/3 arc-second MHW Tsunami Inundation DEM:
www9: National Geophysical Data Center (NGDC) GEODAS:
www10: NOAA Tsunami Inundation Digital Elevation Models (DEMs):
http://www11: SAMRAI:
www12: USGS source for 27 February 2010 earthquake:
www13: Webpage for this paper:
Recommend this journal

Email your librarian or administrator to recommend adding this journal to your organisation's collection.

Acta Numerica
  • ISSN: 0962-4929
  • EISSN: 1474-0508
  • URL: /core/journals/acta-numerica
Please enter your name
Please enter a valid email address
Who would you like to send this to? *


Full text views

Total number of HTML views: 0
Total number of PDF views: 0 *
Loading metrics...

Abstract views

Total abstract views: 0 *
Loading metrics...

* Views captured on Cambridge Core between <date>. This data will be updated every 24 hours.

Usage data cannot currently be displayed