A conservative spectral method for several two-dimensional nonlinear wave equations

A conservative spectral method is proposed to solve several two-dimensional nonlinear wave equations. The conventional fast Fourier transform is used to approximate the spatial derivatives and a three-level difference scheme with a free parameter a is to advance the solution in time. Our time discre tization is semi-implicit in the sense that the linear terms are treated im plicitly while the nonlinear terms are evaluated only by previous time leve ls, and thus treated explicitly. However, the cost of the algorithm is no g reater than that of a fully explicit method because the linear boundary val ue problem that must be solved at each time step is almost trivial in a spe ctral spatial discretization. A linear stability analysis shows that the me thod leads to a less restrictive stability condition than the corresponding explicit one. The method is conservative and the ratio of the numerical di spersion to the physical dispersion is of the order O (Delta t(2)), Applica tions of our method to the Kadomtsev-Petviashvili and the Zakharov-Kuznetso v equations exhibit excellent results. (C) 1999 Academic Press.