Nonlinear irrotational water waves over variable bathymetry. The Hamiltonian approach with a new efficient representation of the dirichlet to neumann operator

A new Hamiltonian formulation for the non-linear water-wave problem over variable bathymetry is presented. It consists of two evolution equations closed by a time-independent, coupled-mode system of horizontal second order linear partial differential equations. The numerical solution of the latter system shows very good accuracy and convergence properties for domains with very steep boundaries. The efficiency of this formulation for solving the nonlinear water wave problem is demonstrated by comparisons of computations against the case of the classical Beji-Battjes experiment, where non-linearity and dispersion are significant.