In this paper, an efficient finite difference scheme for solving the improved Boussinesq equations as derived by Beji and Nodanka(1996) is proposed, The alternate direction iterative method combined with an efficient predictor-corrector scheme are adopted for the numerical solution of the governing differential equations. The proposed numerical scheme is verified by three test cases where experimental data are available lbr comparison. The first case is wave focusing by bottom topography as studied by Whalin (1971). The second case is wave dill? action around a single breakwater by Briggs et al. (1995). The third case is wave diffraction over a shoal as reported by Briggs (1987) and Vincent and Briggs (1989). Numerical results agree very well with the corresponding experimental data in the cases.
The Boussinesq-type equations, which are weakly non-linear and dispersive, are able to give a relatively accurate description of wave transformation in coastal waters. The first such set of equations for nonuniform water depth was derived by Peregrine(1967). These twodimensional equations are obtained by integrating the three-dimensional continuity and momentum equations over the water depth. The reduction of a three-dimensional problem to an equivalent two-dimensional one leads to a drastic savings in computation cost, making the solution of the wave transformation problem tractable. However, these standard Boussinesq equations are inapplicable to intermediate waters because the accuracy of the linear dispersion relation deteriorates with increasing water depth. To extend the range of applicability of the standard Boussinesq equations to deeper waters, alternative forms have been rederivcd by several investigators. Madsen, Murray and Sorensen (1991) presented a new form of equations in terms of the depth-integrated horizontal velocity with improved linear shoaling and dispersive characteristics and Madsen and Sorensen(1992) rederived the new Boussinesq equations for slowing varying bathymetry.
