A local meshless method for solving 2D time-dependent Schrodinger equations

This paper deals with the numerical solutions of the 2D time dependent Schr¨odinger equations by using a local strong form meshless method. The time variable is discretized by a finite difference scheme. Then, in the resultant elliptic type PDEs, special variable is discretized with a local radial basis function (RBF) methods for which the PDE operator is also imposed in the local matrices. Despite the global collocation approaches, dividing the global collocation domain into many local subdomains, the stability of the method increases. Furthermore, because of the use of strong form equation and collocation approach, which does not need integration, and since in the matrix operations the matrices are of small size, computational cost decreases. An iterative approach is proposed to deal with the nonlinear term. Two linear and two nonlinear test problems with known exact solutions are considered and then, the simulation to a nonlinear problem with unknown solution and periodic boundary conditions is also presented and the results reveal that the method is efficient.