Journal of Scientific Computing

, Volume 48, Issue 1–3, pp 105–116 | Cite as

Numerical Simulation of Strongly Nonlinear and Dispersive Waves Using a Green–Naghdi Model

  • F. Chazel
  • D. Lannes
  • F. Marche


We investigate here the ability of a Green–Naghdi model to reproduce strongly nonlinear and dispersive wave propagation. We test in particular the behavior of the new hybrid finite-volume and finite-difference splitting approach recently developed by the authors and collaborators on the challenging benchmark of waves propagating over a submerged bar. Such a configuration requires a model with very good dispersive properties, because of the high-order harmonics generated by topography-induced nonlinear interactions. We thus depart from the aforementioned work and choose to use a new Green–Naghdi system with improved frequency dispersion characteristics. The absence of dry areas also allows us to improve the treatment of the hyperbolic part of the equations. This leads to very satisfying results for the demanding benchmarks under consideration.


Green–Naghdi equations Splitting technique Hybrid method Hyperbolic systems Highorder well-balanced scheme WENO reconstruction Submerged bar Dispersive waves Nonlinear interactions 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Alvarez-Samaniego, B., Lannes, D.: Large time existence for 3D water-waves and asymptotics. Invent. Math. 171(3), 485–541 (2007) CrossRefMathSciNetGoogle Scholar
  2. 2.
    Audusse, E., Bouchut, F., Bristeau, M.-O., Klein, R., Perthame, B.: A fast and stable well-balanced scheme with hydrostatic reconstruction for shallow water flows. J. Comput. Phys. 25(6), 2050–2065 (2004) MATHMathSciNetGoogle Scholar
  3. 3.
    Beji, S., Battjes, J.A.: Experimental investigation of wave propagation over a bar. Coast. Eng. 19, 151–162 (1993) CrossRefGoogle Scholar
  4. 4.
    Berthon, C., Marche, F.: A positive preserving high order VFRoe scheme for shallow water equations: a class of relaxation schemes. SIAM J. Sci. Comput. 30(5), 2587–2612 (2008) MATHCrossRefMathSciNetGoogle Scholar
  5. 5.
    Bonneton, P., Chazel, F., Lannes, D., Marche, F., Tissier, M.: A splitting approach for the fully nonlinear and weakly dispersive Green–Naghdi model (submitted) Google Scholar
  6. 6.
    Carter, J.D., Cienfuegos, R.: Solitary and cnoidal wave solutions of the Serre equations and their stability. Phys. Fluids (2010, submitted) Google Scholar
  7. 7.
    Chazel, F., Benoit, M., Ern, A., Piperno, S.: A double-layer Boussinesq-type model for highly nonlinear and dispersive waves. Proc. R. Soc. Lond. A 465, 2319–2346 (2009) MATHCrossRefMathSciNetGoogle Scholar
  8. 8.
    Cienfuegos, R., Barthelemy, E., Bonneton, P.: A fourth-order compact finite volume scheme for fully nonlinear and weakly dispersive Boussinesq-type equations. Part I: Model development and analysis. Int. J. Numer. Methods Fluids 56, 1217–1253 (2006) CrossRefMathSciNetGoogle Scholar
  9. 9.
    Cienfuegos, R., Barthelemy, E., Bonneton, P.: A fourth-order compact finite volume scheme for fully nonlinear and weakly dispersive Boussinesq-type equations. Part II: Boundary conditions and validations. Int. J. Numer. Methods Fluids 53, 1423–1455 (2007) MATHCrossRefMathSciNetGoogle Scholar
  10. 10.
    Dingemans, M.W.: Comparison of computations with Boussinesq-like models and laboratory measurements. Report H-1684.12, 32, Delft Hydraulics (1994) Google Scholar
  11. 11.
    Godlewski, E., Raviart, P.-A.: Numerical Approximation of Hyperbolic Systems of Conservation Laws. Applied Mathematical Sciences, vol. 118, Springer, Berlin (1996) MATHGoogle Scholar
  12. 12.
    Green, A.E., Naghdi, P.M.: A derivation of equations for wave propagation in water of variable depth. J. Fluid Mech. 78(2), 237–246 (1976) MATHCrossRefGoogle Scholar
  13. 13.
    Jiang, G., Shu, C.-W. Efficient implementation of weighted ENO schemes. J. Comput. Phys. 126, 202–228 (1996) MATHCrossRefMathSciNetGoogle Scholar
  14. 14.
    Shi, J., Hu, C., Shu, C.-W.: A technique of treating negative weights in WENO schemes. J. Comput. Phys. 175, 108–127 (2002) MATHCrossRefGoogle Scholar
  15. 15.
    Lannes, D., Bonneton, P.: Derivation of asymptotic two-dimensional time-dependent equations for surface water wave propagation. Phys. Fluids 21, 016601 (2009) CrossRefGoogle Scholar
  16. 16.
    Le Métayer, O., Gavrilyuk, S., Hank, S.: A numerical scheme for the Green–Naghdi model. J. Comput. Phys. 229(6), 2034–2045 (2010) MATHCrossRefMathSciNetGoogle Scholar
  17. 17.
    Noelle, S., Pankratz, N., Puppo, G., Natvig, J. Well-balanced finite volume schemes of arbitrary order of accuracy for shallow water flows. J. Comput. Phys. 213, 47–499 (2006) CrossRefMathSciNetGoogle Scholar
  18. 18.
    Nwogu, O.G.: An alternative form of the Boussinesq equations for nearshore wave propagation. J. Waterw. Port Coast. Ocean Eng. 119(6), 618–638 (1993) CrossRefGoogle Scholar
  19. 19.
    Seabra-Santos, F.J., Renouard, D.P., Temperville, A.M.: Numerical and experimental study of the transformation of a solitary wave over a shelf or isolated obstacle. J. Fluid Mech. 176, 117–134 (1987) CrossRefGoogle Scholar
  20. 20.
    Su, C.H., Gardner, C.S.: Korteweg-de Vries equation and generalizations. III. Derivation of the Korteweg–de Vries equation and Burgers equation. J. Math. Phys. 10(3), 536–539 (1969) MATHCrossRefMathSciNetGoogle Scholar
  21. 21.
    Wei, G., Kirby, J.T., Grilli, S.T., Subramanya, R.: A fully nonlinear Boussinesq model for surface waves. Part 1. Highly nonlinear unsteady waves. J. Fluid Mech. 294, 71–92 (1995) MATHCrossRefMathSciNetGoogle Scholar
  22. 22.
    Woo, S.-B., Liu, P.L.-F.: A Petrov–Galerkin finite element model for one-dimensional fully nonlinear and weakly dispersive wave propagation. Int. J. Numer. Methods Eng. 37, 541–575 (2001) MATHCrossRefMathSciNetGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC 2010

Authors and Affiliations

  1. 1.Université de ToulouseUPS/INSA, IMT, CNRS UMR 5219ToulouseFrance
  2. 2.DMAEcole Normale Supérieure, et CNRS UMR 8553ParisFrance
  3. 3.I3MUniv. Montpellier 2 et CNRS UMR 5149MontpellierFrance

Personalised recommendations