Advertisement

Modeling Hazardous, Free-Surface Geophysical Flows with Depth-Averaged Hyperbolic Systems and Adaptive Numerical Methods

  • D. L. GeorgeEmail author
Conference paper
Part of the The IMA Volumes in Mathematics and its Applications book series (IMA, volume 156)

Abstract

The mathematical modeling and numerical simulation of gravity-driven, free-surface geophysical flows—such as tsunamis, water floods, and debris flows—are described. These shallow flows are often modeled with two-dimensional depth-averaged equations that possess similar mathematical structure: they are usually nonconservative hyperbolic systems with source terms. Some numerical challenges presented by these systems, particularly with regard to features common to depth-averaged models for flow over topography, are highlighted. The open-source software package GeoClaw incorporates numerical algorithms designed to tackle many of the common difficulties presented by depth-averaged models, particularly for large multiscale problems featuring inundation influenced by topography. Some simulation results for tsunamis, floods, and debris flows highlight the capabilities of GeoClaw.

Keywords

Debris Flow Riemann Problem Shallow Water Equation Riemann Solution Adaptive Mesh Refinement 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Notes

Acknowledgements

The work described in this paper has resulted from an active collaboration of researchers in mathematics, computer science, and the geosciences. In particular, Randall LeVeque at the University of Washington, Seattle; Marsha Berger at the Courant Institute, NYU; and Richard Iverson and Roger Denlinger at the U.S. Geological Survey, Vancouver, WA. GeoClaw software development for future applications continues on an active basis, thanks to contributions from Kyle Mandli at the University of Texas, Austin; Dave Yuen at the University of Minnesota; and many others.

References

  1. 1.
    J.R. Mehelcic, J.C. Crittenden, M.J. Small, D.R. Shonnard, D.R. Hokanson, Q. Zhang, H. Chen, S.A. Sorby, V.U. James, J.W. Sutherland, J.L. Schnoor, Environ. Sci. Technol. 37, 5314 (2003)CrossRefGoogle Scholar
  2. 2.
    J.J. Stoker, Water Waves: The Mathematical Theory with Applications (Interscience Publishers, New York, 1957)zbMATHGoogle Scholar
  3. 3.
    G. Whitham, Linear and Nonlinear Waves (John Wiley and Sons, New York, 1974)zbMATHGoogle Scholar
  4. 4.
    D.T. Resio, J.J. Westerink, Physics Today pp. 33–38 (2008)Google Scholar
  5. 5.
    K.T. Mandli, Finite volume methods for the multilayer shallow water equations with applications to storm surges. Ph.D. thesis, University of Washington, Seattle, WA (2011)Google Scholar
  6. 6.
    R. Abgrall, S. Karni, SIAM Journal on Scientific Computing 31(3), 1603 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    F. Bouchut, V. Zeitlin, Discrete and continuous dynamical systems-series B pp. 739–758 (2010)Google Scholar
  8. 8.
    E.B. Pitman, L. Le, Phil. Trans. R. Soc. A 363, 1573 (2005)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    D.L. George, R.M. Iverson, in The 5th intl. conf. on debris-flow hazards, ed. by R. Genevois, D. Hamilton, A. Prestininzi (Italian Journal of Engineering, Geology and Environment, Padova, Italy, 2011), pp. 415–424Google Scholar
  10. 10.
    R.J. LeVeque, Finite Volume Methods For Hyperbolic Problems. Texts in Applied Mathematics (Cambridge University Press, Cambridge, 2002)Google Scholar
  11. 11.
    G. Dal Maso, P.G. LeFloch, F. Murat., J. Math. Pures Appl. 74, 483 (1995)Google Scholar
  12. 12.
    L. Gosse, Math. Comp. 71, 553 (2002)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    M.J. Castro-Díaz, T. Chacón Rebollo, E.D. Fernández-Nieto, C. Parés., SIAM J. Sci. Comput. 29, 1093 (2007)Google Scholar
  14. 14.
    R.J. LeVeque, Journal of Scientific Computing 48, 209 (2010)MathSciNetCrossRefGoogle Scholar
  15. 15.
    D.L. George, J. Comput. Phys. 227(6), 3089 (2008). DOI 10.1016/j.jcp.2007.10.027MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    V.T. Chow, Open Channel Hydraulics (McGraw-Hill, 1959)Google Scholar
  17. 17.
    D.L. George, Finite volume methods and adaptive refinement for tsunami propagation and inundation. Ph.D. thesis, University of Washington (2006)Google Scholar
  18. 18.
    D.L. George, Int. J. Numer. Meth. Fluids 66(8), 939 (2011). DOI 10.1002/fld.2298CrossRefGoogle Scholar
  19. 19.
    A.E. Green, P. Naghdi, Journal of Fluid Mech. 78, 237 (1976)CrossRefzbMATHGoogle Scholar
  20. 20.
    P.K. Stansby, J.G. Zhou, Int. J. Numer. Meth. Fluids 28, 541 (1998)CrossRefzbMATHGoogle Scholar
  21. 21.
    Z. Li, B. Johns, Int. J. Numer. Meth. Fluids 35, 299 (2001)CrossRefzbMATHGoogle Scholar
  22. 22.
    J. Sainte-Marie, M. Bristeau, DCDS (B) 10(4), 733 (2008)Google Scholar
  23. 23.
    Y. Yamazaki, Z. Kowalik, K. Cheung, Int. J. Numer. Meth. Fluids 61, 473 (2008)MathSciNetCrossRefGoogle Scholar
  24. 24.
    E. Audusse, M. Bristeau, B. Perthame, J. Sainte-Marie, Mathematical Modelling and Numerical Analysis (2009)Google Scholar
  25. 25.
    M.J. Castro-Díaz, E.D. Fernández-Nieto, J.M. González-Vida, C. Parés., Journal of Scientific Computing 48, 16 (2010)Google Scholar
  26. 26.
    T. Takahashi, Annual review of fluid mechanics 13, 57 (1981)CrossRefGoogle Scholar
  27. 27.
    R.M. Iverson, Rev. Geophys. 35(3), 245 (1997)CrossRefGoogle Scholar
  28. 28.
    S.B. Savage, K. Hutter, J. Fluid Mech. 199, 177 (1989)MathSciNetCrossRefzbMATHGoogle Scholar
  29. 29.
    S.B. Savage, K. Hutter, Acta Mech. 86, 201 (1991)MathSciNetCrossRefzbMATHGoogle Scholar
  30. 30.
    A.M. Johnson, Physical Processes in Geology (W. H. Freeman, 1970)Google Scholar
  31. 31.
    R.M. Iverson, R.P. Denlinger, J. Geophys. Res. 106(B1), 537 (2001)CrossRefGoogle Scholar
  32. 32.
    R.P. Denlinger, R.M. Iverson, J. Geophys. Res. 109, F01014 (2004). DOI10.1029/2003 JF000085Google Scholar
  33. 33.
    J. Kowalski, Two-phase modeling of debris flows. Ph.D. thesis, ETH Zurich (2008)Google Scholar
  34. 34.
    M. Pelanti, F. Bouchut, A. Mangeney-Castelnau, J.P. Vilotte., in Hyperbolic Problems: Theory, Numerics, Applications., ed. by S. Benzoni-Gavage, D. Serre (Springer, 2008). Proc. 11th Intl. Conf. on Hyperbolic Problems, Lyon France, July 2006.Google Scholar
  35. 35.
    M. Pelanti, F. Bouchut, A. Mangeney, Mathematical Modelling and Numerical Analysis 42(5), 851 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  36. 36.
    R.M. Iverson, J. Geophys. Res. 110, F02015 (2005)Google Scholar
  37. 37.
    T.W. Lambe, R.V. Whitman, Soil Mechanics (John Wiley and Sons, 1969)Google Scholar
  38. 38.
    M.J. Castro-Díaz, E.D. Fernández-Nieto, A.M. Ferreiro, Comput. Fluids 37, 299 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  39. 39.
    J. Murillo, J. Burguete, P. Brufau, P. García-Navarro, Int. J. Numer. Meth. Fluids 49, 267 (2005). DOI 10.1002/fld.992CrossRefzbMATHGoogle Scholar
  40. 40.
    T. Morales de Luna, M.J. Castro Díaz, C. Parés, E.D.F. Nieto, Communications in Computational Physics 6, 848 (2009)MathSciNetCrossRefGoogle Scholar
  41. 41.
    P. Lynett, P.L.F. Liu, Proceedings of the Royal Society of London A 458, 2885 (2002)MathSciNetCrossRefzbMATHGoogle Scholar
  42. 42.
    M.J. Berger, D.L. George, R.J. LeVeque, K. Mandli, Advances in Water Resources p. in press (2011). DOI 10.1016/j.advwatres.2011.02.016Google Scholar
  43. 43.
    E.F. Toro, Riemann Solvers and Numerical Methods for Fluid Dynamics (Springer–Verlag, Berlin, 1997)CrossRefzbMATHGoogle Scholar
  44. 44.
    R.J. LeVeque, J. Comput. Phys. 131, 327 (1997)CrossRefzbMATHGoogle Scholar
  45. 45.
    V.V. Titov, C.E. Synolakis, Jounal of Waterways, Ports, Coastal and Ocean Engineering 124(4), 157 (1998)Google Scholar
  46. 46.
    J.M. Gallardo, C. Pares, M. Castro, J. Comput. Phys. 227, 574 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  47. 47.
    F. Marche, P. Bonneton, P. Fabrie, N. Seguin, Int. J. Numer. Meth. Fluids 53, 867 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  48. 48.
    P.L. Roe, J. Comput. Phys. 43, 357 (1981)MathSciNetCrossRefzbMATHGoogle Scholar
  49. 49.
    A. Harten, P.D. Lax, B. van Leer, SIAM Review 25, 235 (1983)CrossRefGoogle Scholar
  50. 50.
    R.J. LeVeque, J. Comput. Phys. 146, 346 (1998)MathSciNetCrossRefzbMATHGoogle Scholar
  51. 51.
    J.F. Colombeau, Multiplication of Distributions: A tool in mathematics, numerical engineering and theoretical physics (Springer Verlag, Berlin, 1992)zbMATHGoogle Scholar
  52. 52.
    T. Gallouët, J.M. Hérard, N. Seguin, Comput. Fluids 32, 479 (2003)MathSciNetCrossRefzbMATHGoogle Scholar
  53. 53.
    M.J. Castro, P.G. LeFloch, M.L. Munoz, C. Parés., J. Comput. Phys. 227, 8107 (2008)Google Scholar
  54. 54.
    J.M. Greenberg, A.Y. LeRoux, SIAM J. Numer. Anal. 33, 1 (1996)MathSciNetCrossRefzbMATHGoogle Scholar
  55. 55.
    P. García-Navarro, M.E. Vázquez-Céndon, Comput. Fluids 29, 17 (2000)Google Scholar
  56. 56.
    L. Gosse, Math. Mod. Meth. Appl. Sci. 11, 339 (2001)MathSciNetCrossRefzbMATHGoogle Scholar
  57. 57.
    F. Bouchut, Nonlinear Stability of Finite Volume Methods for Hyperbolic Conservation Laws and Well-Balanced Schemes for Sources (Birkhäuser Verlag, 2004)Google Scholar
  58. 58.
    E.F. Toro, Shock Capturing Methods for Free Surface Shallow Flows (John Wiley and Sons, Chichester, United Kingdom, 2001)zbMATHGoogle Scholar
  59. 59.
    E. Audusse, M.O. Bristeau, J. Comput. Phys. 206, 311 (2005)MathSciNetCrossRefzbMATHGoogle Scholar
  60. 60.
    A. Kurganov, G. Petrova, Commun. Math. Sci. 5(5), 133 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  61. 61.
    E. Audusse, F. Bouchut, M.O. Bristeau, R. Klein, B. Perthame, SIAM J. Sci. Comput. 25, 2050 (2004)MathSciNetCrossRefzbMATHGoogle Scholar
  62. 62.
    R. Bernetti, V. Titarev, E. Toro, J. Comput. Phys. 227, 3212 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  63. 63.
    M.J. Berger, J. Oliger, J. Comput. Phys. 53, 484 (1984)MathSciNetCrossRefzbMATHGoogle Scholar
  64. 64.
    M.J. Berger, P. Colella, J. Comput. Phys. 82, 64 (1989)CrossRefzbMATHGoogle Scholar
  65. 65.
    M.J. Berger, R.J. LeVeque, SIAM J. Numer. Anal. 35, 346 (1998)MathSciNetGoogle Scholar
  66. 66.
    R.J. LeVeque, D.L. George, M.J. Berger, Acta Numerica 20, 211 (2011). DOI10.1017/S09 62492911000043Google Scholar
  67. 67.
    J.M. Hervouet, A. Petitjean, J. Hydraul. Res. 37, 777 (1999).CrossRefGoogle Scholar
  68. 68.
    A. Valiani, V. Caleffi, A. Zanni, J. Hydraul. Eng. 128(5), 460 (2002)CrossRefGoogle Scholar
  69. 69.
    USBOR, Arch dam failures: Malpasset and St. Francis. Tech. rep., Unites States Bureau of Reclamation (unknown)Google Scholar
  70. 70.
    D. Pianese, L. Barbiero, J. Hydraul. Eng. 128(5), 941 (2004)CrossRefGoogle Scholar
  71. 71.
    R. Iverson, M. Logan, R. LaHusen, M. Berti, J. Geophys. Res. 115, 1 (2010)Google Scholar
  72. 72.
    R.M. Iverson, M.E. Reid, N.R. Iverson, R.G. LaHusen, M. Logan, J.E. Mann, D.L. Brien, Science 290(5491), 513 (2000)CrossRefGoogle Scholar
  73. 73.
    A. Mangeney, P. Heinrich, R. Roche, Pure appl. Geophys. 157, 1081 (2000)CrossRefGoogle Scholar
  74. 74.
    R.P. Denlinger, M. Iverson, J. Geophys. Res. 106, 553 (2001)CrossRefGoogle Scholar
  75. 75.
    M. Briggs, C. Synolakis, G. Harkins, Proc. of Waves—Physical and Numerical Modeling (1994)Google Scholar
  76. 76.
    V.V. Titov, C.E. Synolakis, Tsunami 93 Proc. IUGG/IOC Intl Tsunami Symp. pp. 627–636 (1993)Google Scholar

Copyright information

© Springer Science+Business Media New York 2013

Authors and Affiliations

  1. 1.U.S. Geological SurveyVancouverUSA

Personalised recommendations