A TAYLOR-GALERKIN METHOD FOR SIMULATING NONLINEAR DISPERSIVE WATER-WAVES

A new numerical scheme for computing the evolution of water waves with a moderate curvature of the free surface, modeled by the dispersive s hallow water equations, is described. The discretization of this syste m of equations is faced with two kinds of numerical difficulties: the nonsymmetric character of the (nonlinear) advection-propagation operat or and the presence of third order mixed derivatives accounting for th e dispersion phenomenon, In this paper it is shown that the Taylor-Gal erkin finite element method can be used to discretize the problem, ens uring second order accuracy both in time and space and guaranteeing at the same time unconditional stability. The properties of the scheme a re investigated by performing a numerical stability analysis of a line arized model of the scalar 1D regularized long wave equation. The prop osed scheme extends straightforwardly to the fully nonlinear 2D system , which is solved here for the first time on arbitrary unstructured me shes. The results of the numerical simulation of a solitary wave overp assing a vertical circular cylinder are presented and discussed in a p hysical perspective. (C) 1998 Academic Press