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Dispersive wave runup on non-uniform shores

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Finite Volumes for Complex Applications VI Problems & Perspectives

Part of the book series: Springer Proceedings in Mathematics ((PROM,volume 4))

Abstract

Historically the finite volume methods have been developed for the numerical integration of conservation laws. In this study we present some recent results on the application of such schemes to dispersive PDEs. Namely, we solve numerically a representative of Boussinesq type equations in view of important applications to the coastal hydrodynamics. Numerical results of the runup of a moderate wave onto a non-uniform beach are presented along with great lines of the employed numerical method (see D. Dutykh et al. (2011) [6] for more details).

MSC2010: 65M08, 76B15

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References

  1. Audusse, E., Bouchut, F., Bristeau, O., Klein, R., Perthame, B.: A fast and stable well-balanced scheme with hydrostatic reconstruction for shallow water flows. SIAM J. of Sc. Comp. 25, 2050–2065 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  2. Benjamin, T., Olver, P.: Hamiltonian structure, symmetries and conservation laws for water waves. J. Fluid Mech 125, 137–185 (1982)

    Article  MathSciNet  MATH  Google Scholar 

  3. Delis, A.I., Katsaounis, T.: Relaxation schemes for the shallow water equations. Int. J. Numer. Meth. Fluids 41, 695–719 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  4. Dutykh, D., Dias, F.: Dissipative Boussinesq equations. C. R. Mecanique 335, 559–583 (2007)

    MATH  Google Scholar 

  5. Dutykh, D., Dias, F.: Water waves generated by a moving bottom. In: A. Kundu (ed.) Tsunami and Nonlinear waves. Springer Verlag (Geo Sc.) (2007)

    Google Scholar 

  6. Dutykh, D., Katsaounis, T., Mitsotakis, D.: Finite volume schemes for dispersive wave propagation and runup. Accepted to Journal of Computational Physics http://hal.archives-ouvertes.fr/hal-00472431/ (2011)

  7. Dutykh, D., Poncet, R., Dias, F.: Complete numerical modelling of tsunami waves: generation, propagation and inundation. Submitted http://arxiv.org/abs/1002.4553 (2010)

  8. Fokas, A.S., Pelloni, B.: Boundary value problems for Boussinesq type systems. Math. Phys. Anal. Geom. 8, 59–96 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  9. Ghidaglia, J.M., Kumbaro, A., Coq, G.L.: On the numerical solution to two fluid models via cell centered finite volume method. Eur. J. Mech. B/Fluids 20, 841–867 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  10. Harten, A., Osher, S.: Uniformly high-order accurate nonscillatory schemes, I. SIAM J. Numer. Anal. 24, 279–309 (1987)

    Article  MathSciNet  MATH  Google Scholar 

  11. Hibberd, S., Peregrine, D.: Surf and run-up on a beach: a uniform bore. J. Fluid Mech. 95, 323–345 (1979)

    Article  MathSciNet  MATH  Google Scholar 

  12. Kervella, Y., Dutykh, D., Dias, F.: Comparison between three-dimensional linear and nonlinear tsunami generation models. Theor. Comput. Fluid Dyn. 21, 245–269 (2007)

    Article  MATH  Google Scholar 

  13. van Leer, B.: Towards the ultimate conservative difference scheme V: a second order sequel to Godunov’ method. J. Comput. Phys. 32, 101–136 (1979)

    Article  Google Scholar 

  14. Peregrine, D.H.: Long waves on a beach. J. Fluid Mech. 27, 815–827 (1967)

    Article  MATH  Google Scholar 

  15. Simon, M.: Wave-energy extraction by a submerged cylindrical resonant duct. Journal of Fluid Mechanics 104, 159–187 (1981)

    Article  MATH  Google Scholar 

  16. Tadepalli, S., Synolakis, C.E.: The run-up of N-waves on sloping beaches. Proc. R. Soc. Lond. A 445, 99–112 (1994)

    Article  MATH  Google Scholar 

  17. Titov, V., González, F.: Implementation and testing of the method of splitting tsunami (MOST) model. Tech. Rep. ERL PMEL-112, Pacific Marine Environmental Laboratory, NOAA (1997)

    Google Scholar 

  18. Titov, V.V., Synolakis, C.E.: Numerical modeling of tidal wave runup. J. Waterway, Port, Coastal, and Ocean Engineering 124, 157–171 (1998)

    Google Scholar 

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Acknowledgements

D. Dutykh acknowledges the support from French Agence Nationale de la Recherche, project MathOcean (Grant ANR-08-BLAN-0301-01) and Ulysses Program of the French Ministry of Foreign Affairs under the project 23725ZA. The work of Th. Katsaounis was partially supported by European Union FP7 program Capacities(Regpot 2009-1), through ACMAC (http://acmac.tem.uoc.gr).

Author information

Authors and Affiliations

  1. LAMA, UMR 5127 CNRS, Université de Savoie, Campus Scientifique, 73376, Le Bourget-du-Lac Cedex, France

    Denys Dutykh

  2. Department of Applied Mathematics, University of Crete, Heraklion, 71409 Greece Inst. of App. and Comp. Math. (IACM), FORTH, Heraklion, 71110, Greece

    Theodoros Katsaounis

  3. IMA, University of Minnesota, MN, Minneapolis, 55455, USA

    Dimitrios Mitsotakis

Authors
  1. Denys Dutykh
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  2. Theodoros Katsaounis
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  3. Dimitrios Mitsotakis
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Corresponding author

Correspondence to Denys Dutykh .

Editor information

Editors and Affiliations

  1. , Faculty of Mechanical Engineering, Czech Technical University, Karlovo náměstí 13, Prague, Czech Republic

    Jaroslav Fořt

  2. , Faculty of Mechanical Engineering, Czech Technical University, Karlovo náměstí 13, Prague, Czech Republic

    Jiří Fürst

  3. , Faculty of Mechanical Engineering, Czech Technical University, Karlovo náměstí 13, Prague, Czech Republic

    Jan Halama

  4. LATP, Laboratoire d'Analyse, Probabilités et Topologie, Université Aix-Marseille, rue Joliot Curie 39, Marseille, 13453, France

    Raphaèle Herbin

  5. LATP, Laboratoire d'Analyse, Probabilités et Topologie, Université Aix-Marseille, rue Joliot Curie 39, Marseille, 13453, France

    Florence Hubert

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Dutykh, D., Katsaounis, T., Mitsotakis, D. (2011). Dispersive wave runup on non-uniform shores. In: Fořt, J., Fürst, J., Halama, J., Herbin, R., Hubert, F. (eds) Finite Volumes for Complex Applications VI Problems & Perspectives. Springer Proceedings in Mathematics, vol 4. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-20671-9_41

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