This paper develops and analyses a finite difference/spectral-Galerkin scheme for the nonlinear fractional Schrödinger equations with Riesz space- and Caputo time-fractional derivatives. The $L1$ finite difference approximation is used for the discretization of the Caputo fractional derivative and the Legendre-Galerkin spectral method is used for the spatial approximation. Additionally, by using a proper form of discrete Grönwall inequality, the scheme is proved to be unconditionally stable and convergent with $2-\text{β}$ accuracy in time and spectral accuracy in space in case of smooth solutions. Finally, some numerical tests are preformed to distinguish the validity of our theoretical results.

本文开发并分析了具有 Riesz 空间和 Caputo 时间分数阶导数的非线性分数阶薛定谔方程的有限差分/谱伽辽金格式。这$升1$有限差分近似用于 Caputo 分数阶导数的离散化，而 Legendre-Galerkin 谱方法用于空间近似。此外，通过使用离散 Grönwall 不等式的适当形式，该方案被证明是无条件稳定的并且收敛于$2-\text{β}$在平滑解的情况下，时间精度和空间光谱精度。最后，进行了一些数值测试，以区分我们的理论结果的有效性。