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 $L^{\infty }$-norm boundness of the numerical solution of the proposed scheme. However, we cannot obtain the $L^{\infty }$-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 $L^{\infty }$-norm boundness of the numerical solution and $L^2$-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.

在本文中，我们打算为具有积分分数拉普拉斯算子的二维空间分数非线性薛定谔（2D SFNS）方程构造一个结构保持差分格式。时间方向通过改进的 Crank-Nicolson 方法离散化，空间变量通过一种新的分数中心差分方法近似。所提出方案的质量和能量守恒以及收敛性得到了严格证明。对于一维 SFNS 方程，收敛严重依赖于${升}^{\infty }$-所提议方案的数值解的范数有界。然而，我们无法获得${升}^{\infty }$-通过对二维 SFNS 方程使用类似过程来计算数值解的范数有界。本文的主要意义之一是我们首先获得${升}^{\infty }$-数值解的范数有界和 ${升}^2$-通过二维 SFNS 方程流行的“截止”函数进行范数误差估计。此外，我们揭示了空间离散化会生成一个块托普利兹系数矩阵，并且随着空间网格网格数M和分数阶数α 的增加，它将是病态的。因此，我们为非线性系统开发了一种线性化迭代算法，以便 Krylov 子空间求解器可以使用合适的预处理器有效地求解它，其中在求解器中应用 2D 快速傅立叶变换 (2D FFT) 以加速矩阵 -矢量积，采用标准正交投影法消除质量和能量的漂移。大量的数值结果报告证实了所提出算法的理论分析和高效率。