Abstract
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
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
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.