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

Abstract

This paper presents two second-order and linear finite element schemes for the multi-dimensional nonlinear time-fractional Schrodinger equation. In the first numerical scheme, we adopt the L2-1σ 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 $\mathcal {F}L2$ -1σ formula to construct the second numerical scheme, which reduces the computational storage and cost. We prove that both the L2-1σ and $\mathcal {F}L2$ -1σ formulas satisfy the three assumptions of the generalized discrete fractional Gronwall 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 $\mathcal {F}L2$ -1σ formula can reduce CPU time significantly compared with the first scheme based on L2-1σ formula.