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Fast high-order compact finite difference schemes are investigated for solving the two-dimensional nonlinear Schrödinger equation with periodic boundary conditions. These schemes are convergent of order  s  in space, where () with second-order temporal accuracy. The discrete conservation laws and convergence of the finite difference schemes are rigorously demonstrated. Thanks to the circulant matrices resulting from spatial discretization, we significantly reduce the computation complexity and storage requirement of the proposed schemes via fast Fourier transform. Numerical examples are presented to show the accuracy and efficiency of these schemes and verify the theoretical analysis. Moreover, we extend the tenth-order scheme to solve the generalized nonlinear Schrödinger equation.

Fast high-accuracy compact conservative difference schemes for solving the nonlinear Schrödinger equation