Fully dispersive nonlinear water wave model in curvilinear coordinates.

*(English)*Zbl 1116.76324Summary: A vertically integrated fully dispersive nonlinear wave model is expressed in curvilinear coordinates with non-orthogonal grids for the simulation of broad-banded nonlinear random water waves in regions of arbitrary geometry. The transformation is performed for both dependent and independent variables, hence an irregular physical domain is converted into a rectangular computational domain with contravariant velocities. Use of contravariant velocity components as dependent variables ensures easy and accurate satisfaction of the wall condition for lateral enclosures surrounding a physical domain, such as a coastal area, channel, or harbor. The numerical scheme is based on finite-difference approximations with staggered grids which results in implicit formulations for the momentum equations and a semi-explicit formulation for the continuity equation. Linear long wave propagation in a channel of varying cross-section and linear random wave propagation in a circular channel are presented as test cases for comparisons with the corresponding analytical solutions. Cnoidal and Stokes waves in a circular channel are also simulated as examples to nonlinear wave propagation within curved walls.

##### MSC:

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

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

65M06 | Finite difference methods for initial value and initial-boundary value problems involving PDEs |

86A05 | Hydrology, hydrography, oceanography |

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\textit{S. Beji} and \textit{K. Nadaoka}, J. Comput. Phys. 198, No. 2, 645--658 (2004; Zbl 1116.76324)

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