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**Numerical simulation of strongly nonlinear and dispersive waves using a Green-Naghdi model.**
*(English)*
Zbl 1419.76454

Summary: 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.

### MSC:

76M12 | Finite volume methods applied to problems in fluid mechanics |

76M20 | Finite difference methods applied to problems in fluid mechanics |

76B15 | Water waves, gravity waves; dispersion and scattering, nonlinear interaction |

### Keywords:

Green-Naghdi equations; splitting technique; hybrid method; hyperbolic systems; highorder well-balanced scheme; WENO reconstruction; submerged bar; dispersive waves; nonlinear interactions
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\textit{F. Chazel} et al., J. Sci. Comput. 48, No. 1--3, 105--116 (2011; Zbl 1419.76454)

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### References:

[1] | Alvarez-Samaniego, B., Lannes, D.: Large time existence for 3D water-waves and asymptotics. Invent. Math. 171(3), 485–541 (2007) · Zbl 1131.76012 |

[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) · Zbl 1133.65308 |

[3] | Beji, S., Battjes, J.A.: Experimental investigation of wave propagation over a bar. Coast. Eng. 19, 151–162 (1993) |

[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) · Zbl 1358.76053 |

[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) · Zbl 1391.76066 |

[6] | Carter, J.D., Cienfuegos, R.: Solitary and cnoidal wave solutions of the Serre equations and their stability. Phys. Fluids (2010, submitted) · Zbl 1258.76047 |

[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) · Zbl 1186.35156 |

[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) · Zbl 1158.76361 |

[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) · Zbl 1370.76090 |

[10] | Dingemans, M.W.: Comparison of computations with Boussinesq-like models and laboratory measurements. Report H-1684.12, 32, Delft Hydraulics (1994) |

[11] | Godlewski, E., Raviart, P.-A.: Numerical Approximation of Hyperbolic Systems of Conservation Laws. Applied Mathematical Sciences, vol. 118, Springer, Berlin (1996) · Zbl 0860.65075 |

[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) · Zbl 0351.76014 |

[13] | Jiang, G., Shu, C.-W. Efficient implementation of weighted ENO schemes. J. Comput. Phys. 126, 202–228 (1996) · Zbl 0877.65065 |

[14] | Shi, J., Hu, C., Shu, C.-W.: A technique of treating negative weights in WENO schemes. J. Comput. Phys. 175, 108–127 (2002) · Zbl 0992.65094 |

[15] | Lannes, D., Bonneton, P.: Derivation of asymptotic two-dimensional time-dependent equations for surface water wave propagation. Phys. Fluids 21, 016601 (2009) · Zbl 1183.76294 |

[16] | Le Métayer, O., Gavrilyuk, S., Hank, S.: A numerical scheme for the Green–Naghdi model. J. Comput. Phys. 229(6), 2034–2045 (2010) · Zbl 1334.74015 |

[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) · Zbl 1088.76037 |

[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) |

[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) |

[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) · Zbl 0283.35020 |

[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) · Zbl 0859.76009 |

[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) · Zbl 1109.76333 |

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