(untitled) (Advances in Applied Mathematics and Mechanics, 2023)
This paper aims to build a new framework of convergence analysis of conservative Fourier pseudo-spectral method for the general nonlinear Schrödinger equation in two dimensions, which is not restricted that the nonlinear term is mere cubic. The new framework of convergence analysis consists of two steps. In the first step, by truncating the nonlinear term into a global Lipschitz function, an alternative numerical method is proposed and proved in a rigorous way to be convergent in the discrete L^2 norm; followed in the second step, the maximum bound of the numerical solution of the alternative numerical method is obtained by using a lifting technique, as implies that the two numerical methods are the same one. Under our framework of convergence analysis, with neither any restriction on the grid ratio nor any requirement of the small initial value, we establish the error estimate of the proposed conservative Fourier pseudo-spectral method, while previous work requires the certain restriction for the focusing case. The error bound is proved to be of O(h^r+τ^2 ) with grid size h and time step τ. In fact, the framework can be used to prove the unconditional convergence of many other Fourier pseudo-spectral methods for solving the nonlinear Schrödinger-type equations. Numerical results are conducted to indicate the accuracy and efficiency of the proposed method, and investigate the effect of the nonlinear term and initial data on the blow-up solution.