A Conservative Spectral Method for Several Two-Dimensional Nonlinear Wave Equations

✍ Scribed by B.-F Feng; T Kawahara; T Mitsui

• Publisher: Elsevier Science
• Year: 1999
• Tongue: English
• Weight: 286 KB
• Volume: 153
• Category: Article
• ISSN: 0021-9991
• DOI: 10.1006/jcph.1999.6286

✦ Synopsis

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 θ is to advance the solution in time. Our time discretization is semi-implicit in the sense that the linear terms are treated implicitly while the nonlinear terms are evaluated only by previous time levels, and thus treated explicitly. However, the cost of the algorithm is no greater than that of a fully explicit method because the linear boundary value problem that must be solved at each time step is almost trivial in a spectral spatial discretization. A linear stability analysis shows that the method leads to a less restrictive stability condition than the corresponding explicit one. The method is conservative and the ratio of the numerical dispersion to the physical dispersion is of the order O( t 2 ). Applications of our method to the Kadomtsev-Petviashvili and the Zakharov-Kuznetsov equations exhibit excellent results.