Two second-order and linear numerical schemes for the multi-dimensional nonlinear time-fractional Schrödinger equation

This paper presents two second-order and linear finite element schemes for the multi-dimensional nonlinear time-fractional Schrödinger equation. In the first numerical scheme, we adopt the L2-1_σ formula to approximate the Caputo derivative. However, this scheme requires storing the numerical solution at all previous time steps. In order to overcome this drawback, we develop the FL2 -1_σ formula to construct the second numerical scheme, which reduces the computational storage and cost. We prove that both the L2-1_σ and FL2 -1_σ formulas satisfy the three assumptions of the generalized discrete fractional Grönwall inequality. Furthermore, combining with the temporal-spatial error splitting argument, we rigorously prove the unconditional stability and optimal error estimates of these two numerical schemes, which do not require any time-step restrictions dependent on the spatial mesh size. Numerical examples in two and three dimensions are given to illustrate our theoretical results and show that the second scheme based on FL2 -1_σ formula can reduce CPU time significantly compared with the first scheme based on L2-1_σ formula.