Nonlinear wave modelling over variable water depth using extended boussinesq equations

Abstract

Numerical modeling of wave-ship interaction in shallow water over variable depth requires an accurate description of diffraction, refraction, reflection, and nonlinear wave-wave interaction. A computer program has been developed to solve time dependent Boussinesq-type hyperbolic long wave equations. The velocity at an arbitrary depth is expanded into an infinite series for the formulation of the extended Boussinesq equations. The numerical stability and dispersion characteristics are improved for increasing water depths. The partial differential equations are solved by using a fifth-order Adams-Bashforth-Moulton time marching multistep finite difference method. The results are compared with a second-order Crank Nicolson finite difference method and a Galerkin finite element method from previously published results. Results for linear and nonlinear waves are also compared with analytical and experimental data. The program will be integrated with a time-domain seakeeping program to simulate wave-ship interaction in coast al waves. The current research contributes higher order time and space discretizations, and generalizes the numerical algorithm for methods of any order given the coefficients for finite difference equations. The research allows for higher-order Boussinesq equations while minimizing the numerical error from the time and space differential approximations.