Finite Analytic Method for Mild-Slope Wave Equation

Abstract

A difference scheme for the mild-slope wave equation that governs the combined diffraction and refraction of nearshore waves is derived based on the finite analytic method. The nine-point scheme expresses the value of the dependent variable at the central point of a rectangular grid element as a linear combination of its values at the surrounding nodes. The coefficients of the linear combination depend on the local wave number as well as the width-to-length ratio of the grid element and are approximated, by Taylor expansion, as the polynomials of the relative mesh size with coefficients being functions of the width-to-length ratio of the grid element. Performance of the scheme is studied by comparing the numerical results with the exact solution for a Dirichlet problem and with the solution by separation of variables for the forced oscillation in a square basin. The scheme is also applied to the computation of wave diffraction by the spherical shoal in a wave channel, on which carefully measured data in laboratory are available for comparison.