Unconditional Convergence of Linearized TL1 Difference Methods for a Time-Fractional Coupled Nonlinear Schrödinger System

This paper presents a transformed L1 (TL1) finite difference method for the time-fractional coupled nonlinear Schrödinger system. Unconditionally optimal $L^2$ error estimates of the fully discrete scheme are obtained. The convergence results indicate that the method has an order of $2$ in the spatial direction and an order of $2 − α$ in the temporal direction. The error estimates hold without any spatial-temporal stepsize restriction. Such convergence results are obtained by applying a novel discrete fractional Grönwall inequality and the corresponding Sobolev embedding theorems. Numerical experiments for both two-dimensional and three-dimensional models are carried out to confirm our theoretical findings.