A second-order finite difference scheme for the multi-dimensional nonlinear time-fractional Schrödinger equation

This paper is concerned with a linearized second-order finite difference scheme for solving the nonlinear time-fractional Schrödinger equation in d ( d = 1,2,3) dimensions. Under a weak assumption on the nonlinearity, the optimal error estimate of the numerical solution is established without any restriction on the grid ratio. Besides the standard energy method, the key tools for analysis include the mathematical induction method, several inverse Sobolev inequalities, and a discrete fractional Gronwall-type inequality. The convergence rate of the proposed scheme is of O ( τ 2 + h 2 ) with time step τ and mesh size h . Numerical results are carried out to confirm the theoretical analysis.