In this paper, a linearized semi-implicit finite difference scheme is proposed for solving the two-dimensional (2D) space fractional nonlinear Schr\"{o}dinger equation (SFNSE).The scheme has the property of mass and energy conservation on the discrete level, with an unconditional stability and a second order accuracy for both time and spatial variables. The main contribution of this paper is an optimal pointwise error estimate for the 2D SFNSE, which is rigorously established and proved for the first time. Moreover, a novel technique is proposed for dealing with the nonlinear term in the equation, which plays an essential role in the error estimation. Finally, the numerical results confirm well with the theoretical findings.

翻译：在本文中,为解决二维(2D)空间分数非线性Schr\"{o}dinger等式(SFNSE),提出了线性半隐含的半隐性差别方案。这个方案在离散水平上具有质量和节能特性,对时间和空间变量都具有无条件的稳定性和第二顺序的精确度。本文的主要贡献是对2D SFNSE进行最佳的点向误差估计,这是首次严格建立和证明的。此外,还提出了处理该等式中非线性术语的新技术,该术语在误差估计中起着至关重要的作用。最后,数字结果与理论结论完全吻合。