Abstract
In this paper, we study the numerical methods for solving the time-fractional Schrodinger equation (TFSE) with Caputo or Riemann-Liouville fractional derivative. The numerical schemes are implemented by using the L1 scheme in time direction and Fourier-Galerkin/Legendre-Galerkin spectral methods in spatial variable. We prove that the two schemes are unconditionally stable and numerical solutions converge with the order O(Delta t2-alpha+N-s+N-m), where alpha is the order of the fractional derivative, Delta t, N are the step of time and polynomial degree, respectively, m, s are the regularity of u and V. Several numerical results are performed to confirm the theoretical analysis.