Linearized Crank–Nicolson scheme for the nonlinear time–space fractional Schrödinger equations

Abstract

In this paper, a Crank–Nicolson difference scheme is first derived for solving the nonlinear time–space fractional Schrödinger equations. The truncation error and stability of the scheme are discussed in detail. The existence of the numerical solution is shown by the Brouwer fixed point theorem. For improving the calculating efficiency, a three-level linearized difference scheme is also proposed and analyzed. Both schemes are subsequently extended to the nonlinear coupled equations, and some similar results are given and proved. Several numerical experiments are included to verify the accuracy and efficiency of the two types of schemes, and comparison with the related work is presented. • The single and coupled time–space fractional Schrödinger equations are studied. • The Crank–Nicolson scheme and linearized scheme are derived. • The existence, truncation error and unconditional stability of both schemes are proved. • The accuracy and robustness of both schemes are validated numerically.