An efficient numerical technique for solution of two-dimensional cubic nonlinear Schrödinger equation with error analysis

In this paper, the spectral meshless radial point interpolation (SMRPI) technique is applied to the solution of two-dimensional cubic nonlinear Schrödinger equations. Firstly, we obtain a time discrete scheme by approximating time derivative via a finite difference formula, then we use the SMRPI approach to approximate the spatial derivatives. This method is based on a combination of meshless methods and spectral collocation techniques. The point interpolation method with the help of radial basis functions is used to construct shape functions which act as basis functions in frame of SMRPI. In the current work, the thin plate splines (TPS) are used as the basis functions and in order to eliminate the nonlinearity, a simple predictor-corrector (P-C) scheme is performed. We prove that the time discrete scheme is unconditionally stable and convergent in time variable using the energy method. We show that convergence order of the time discrete scheme is O(δt). The aim of this paper is to show that the SMRPI method is suitable for the treatment of the nonlinear Schrödinger equations. Also, the SMRPI has less computational complexity than the other methods that have already solved this problem. The results of numerical experiments are compared with analytical solution to confirm the accuracy and efficiency of the presented scheme.