In this paper, a linearized semi-implicit finite difference scheme is proposed for solving the two-dimensional (2D) space fractional nonlinear Schrödinger equation (SFNSE). The scheme has the property of mass and energy conservation at 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 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.

中文翻译：

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二维空间分数阶非线性薛定谔方程的具有最优逐点误差估计的保守差分方案

在本文中，提出了一种用于求解二维（2D）空间分数阶非线性薛定谔方程（SFNSE）的线性化半隐式有限差分格式。该方案具有离散水平的质量和能量守恒性质，具有无条件稳定性和时间和空间变量的二阶精度。本文的主要贡献是首次严格建立的 2D SFNSE 的最优逐点误差估计。此外，还提出了一种新的技术来处理方程中的非线性项，它在误差估计中起着至关重要的作用。最后，数值结果很好地证实了理论结果。