Legendre-tau-Galerkin and spectral collocation method for nonlinear evolution equations

摘要

A Legendre-tau-Galerkin method is developed for nonlinear evolution problems and its multiple interval form is also considered. The Legendre tau method is applied in time and the Legendre/Chebyshev-Gauss-Lobatto points are adopted to deal with the nonlinear term. By taking appropriate basis functions, it leads to a simple discrete equation. The proposed method enables us to derive optimal error estimates in L~2-norm for the Legendre collocation under the two kinds of Lipschitz conditions, respectively. Our method is also applied to the numerical solutions of some nonlinear partial differential equations by using the Legendre Galerkin and Chebyshev collocation in spatial discretization. Numerical examples are given to show the efficiency of the methods.