A spatial sixth-order alternating direction implicit method for two-dimensional cubic nonlinear Schrödinger equations

Based on the combined compact difference scheme, an alternating direction implicit method is proposed for solving two-dimensional cubic nonlinear Schrödinger equations. The proposed method is sixth-order accurate in space and second-order accurate in time. The linear Fourier analysis method is exploited to study the stability of the proposed method. The efficiency and accuracy of the proposed method are tested numerically. The common solution pattern of the nonlinear Schrödinger equation is also illustrated using relevant examples known in the literature.