A New Framework of Convergence Analysis for Solving the General Nonlinear Schrödinger Equation Using the Fourier Pseudo-Spectral Method in Two Dimensions (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, bytruncating the nonlinear term into a global Lipschitz function, an alternative numericalmethod 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 ofthe alternative numerical method is obtained by using a lifting technique, as impliesthat 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 ofthe small initial value, we establish the error estimate of the proposed conservativeFourier pseudo-spectral method, while previous work requires the certain restrictionfor the focusing case. The error bound is proved to be of $\mathcal{O}(h^r+τ^2)$ 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 nonlinearSchrödinger-type equations. Numerical results are conducted to indicate the accuracy and efficiency of the proposed method, and investigate the effect of the nonlinear termand initial data on the blow-up solution.