Hybrid-Spectral Model for Fully Nonlinear Numerical Wave Tank∗

This abstract is concerned with the development of an efficient and scalable hybrid-spectral model for a fully nonlinear numerical wave tank with finite dimensions, which is used for simulation of nonlinear free surface waves generated by a moving wavemaker. The approach pursued here is based on the OceanWave3D strategy established in [1, 6, 7], replacing the horisontal high order finite difference approximations used in OceanWave3D with a Fourier Collocation method based on cosine modes. This is combined with a modal Chebyshev Tau method for the potential in the vertical, which by applying an efficient strategy for solving the discrete Laplace equation yields a new attractive and accurate Dirichletto-Neumann operator, recently established in [4]. The application of hybrid-spectral methods in the context of the OceanWave3D strategy is motivated by the comparative study of two serial implementations of nonlinear water wave models carried out in [5], which demonstrated that the high-order finite difference model (OceanWave3D) requires approximately an order of magnitude larger computational effort than the high order spectral model (HOS) in order to solve highly nonlinear water wave problems to the same level of accuracy. Compared to the HOS model, the main advantage of OceanWave3D is a relatively straight-forward introduction of nonrectangular geometry, in particular varying bottom bathymetry and the generation of waves by a moving wavemaker. These attractive properties, associated with the OceanWave3D solution strategy, are retained in the presented hybrid-spectral method. In this abstract the governing equations for the fully nonlinear free surface potential wave problem are derived in curvlinear coordinates on a fixed computational domain, which allows the fully nonlinear wavemaker condition to be satisfied directly. The hybrid-spectral discretization strategy and iterative solution of the resulting discrete Laplace problem are detailed, and it is described how the wavemaker can be modelled by the introduction of additional potentials following the line of [2], which with the present discretization strategy can be obtained essentially free of cost.