On convergence of a structure preserving difference scheme for two-dimensional space-fractional nonlinear Schrödinger equation and its fast implementation

Abstract

In this paper we intend to construct a structure preserving difference scheme for two-dimensional space-fractional nonlinear Schrödinger (2D SFNS) equation with the integral fractional Laplacian. The temporal direction is discretized by the modified Crank-Nicolson method, and the spatial variable is approximated by a novel fractional central difference method. The mass and energy conservations and the convergence are rigorously proved for the proposed scheme. For 1D SFNS equation, the convergence relies heavily on the L∞-norm boundness of the numerical solution of the proposed scheme. However, we cannot obtain the L∞-norm boundness of the numerical solution by using the similar process for the 2D SFNS equation. One of the major significance of this paper is that we first obtain the L∞-norm boundness of the numerical solution and L2-norm error estimate via the popular “cut-off” function for the 2D SFNS equation. Further, we reveal that the spatial discretization generates a block-Toeplitz coefficient matrix, and it will be ill-conditioned as the spatial grid mesh number M and the fractional order α increase. Thus, we exploit an linearized iteration algorithm for the nonlinear system, so that it can be efficiently solved by the Krylov subspace solver with a suitable preconditioner, where the 2D fast Fourier transform (2D FFT) is applied in the solver to accelerate the matrix-vector product, and the standard orthogonal projection approach is used to eliminate the drift of mass and energy. Extensive numerical results are reported to confirm the theoretical analysis and high efficiency of the proposed algorithm.