Numerical analysis for time-fractional Schrödinger equation on two space dimensions

In this paper, we study the numerical methods for solving the time-fractional Schrödinger 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(Δt2−α+N−s+N−m)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\mathcal{O}( \Delta t^{2-\alpha }+N^{-s}+ N^{-m})$\end{document}, where α is the order of the fractional derivative, Δt, 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.