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On the propagation of nonlinear water waves in a three-dimensional numerical wave flume using the generalized finite difference method. (English) Zbl 1464.76123
Summary: Nonlinear water waves are common physical phenomena in the field of coastal and ocean engineering, which plays a critical role in the investigation of hydrodynamics regarding offshore and deep-water structures. In the present study, a three-dimensional (3D) numerical wave flume (NWF) is constructed to simulate the propagation of nonlinear water waves. On the basis of potential flow theory, the second-order Runge-Kutta method (RKM2) combining with a semi-Lagrangian approach is carried out to discretize the temporal variable of the 3D Laplace’s equation. For the spatial variables, the generalized finite difference method (GFDM) is adopted to solve the governing equations for the deformable computational domain at each time step. The upstream condition is considered as a wave-making boundary with imposing horizontal velocity while the downstream condition as a wave-absorbing boundary with a pre-defined sponge layer to deal with the phenomenon of wave reflection. Three numerical examples are investigated and discussed in detail to validate the accuracy and stability of the developed 3D GFDM-based NWF. The results show that the newly-proposed numerical method has good performance in the prediction of the dynamic evolution of nonlinear water waves, and suggests that the novel 3D “RKM2-GFDM” meshless scheme can be employed to further investigate more complicated hydrodynamic problems in practical applications.
##### MSC:
 76M20 Finite difference methods applied to problems in fluid mechanics 65M06 Finite difference methods for initial value and initial-boundary value problems involving PDEs 76B15 Water waves, gravity waves; dispersion and scattering, nonlinear interaction
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