A fast method for nonlinear three-dimensional free-surface waves

An efficient numerical model for solving fully nonlinear potential flow equations with a free surface is presented. Like the code that was developed by Grilli et al . (Grilli et al . 2001 Int. J. Numer. Methods Fluids 35 , 829–867), it uses a high-order three-dimensional boundary-element method combined with mixed Eulerian–Lagrangian time updating, based on second-order explicit Taylor expansions with adaptive time-steps. Such methods are known to be accurate but expensive. The efficiency of the code has been greatly improved by introducing the fast multipole algorithm. By replacing every matrix–vector product of the iterative solver and avoiding the building of the influence matrix, this algorithm reduces the computing complexity from to nearly , where N is the number of nodes on the boundary. The performance of the method is illustrated by the example of the overturning of a solitary wave over a three-dimensional sloping bottom. For this test case, the accelerated method is indeed much faster than the former one, even for quite coarse grids. For instance, a reduction of the complexity by a factor six is obtained for N =6022, for the same global accuracy. The acceleration of the code allows the study of more complex physical problems and several examples are presented.