A Compact Split-step Finite Difference Method for Solving the Nonlinear Schrödinger Equation

Abstract

The nonlinear Schrödinger equation arises from quantum mechanics and is extensively used in many fields of science and engineering. Thus, it is important to construct the high-order and stable numerical scheme of the Schrödinger equation. To solve the high-order and stable numerical solution of the nonlinear Schrödinger equation, the compact split-step finite difference method and the local one-dimensional method are combined in this paper. To attain high-order accuracy in time and space, the 4-order compact finite difference in space discretization is combined with the L-stable Simpson method in time discretization. Therefore, a scheme with 4-order accuracy in space and 3-order accuracy in time is obtained, and the stability of the scheme is analyzed. Finally, numerical results manifest that the devised scheme can supply accurate and stable results to the nonlinear Schrödinger equation.