Efficient semi-implicit compact finite difference scheme for nonlinear Schrödinger equations on unbounded domain

The compact finite difference scheme is designed to numerically solve the nonlinear Schrodinger equation and coupled nonlinear Schrodinger equations on an unbounded domain in this paper. The original problem on an unbounded domain is reduced to an initial boundary value problem defined on a bounded computational domain by applying the artificial boundary method. Then, the reduced problem on the bounded computational domain is solved by an efficient semi-implicit compact finite difference scheme, which is a fourth-order scheme with respect to spatial variable. The scheme efficiently avoids the time-consuming iteration procedure necessary for the nonlinear scheme and thus is relatively time-saving. Finally, the stability of the proposed scheme is rigorously analyzed. Numerical examples are given to illustrate the accuracy and effectiveness of the proposed method.