Linearized Crank-Nicolson scheme for the nonlinear time-space fractional Schrödinger equations. (English) Zbl 1419.65027
Summary: 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.
MSC:
| 65M06 | Finite difference methods for initial value and initial-boundary value problems involving PDEs |
| 35Q41 | Time-dependent Schrödinger equations and Dirac equations |
| 65M12 | Stability and convergence of numerical methods for initial value and initial-boundary value problems involving PDEs |
| 35R11 | Fractional partial differential equations |
Keywords:
fractional Schrödinger equation; time-space fractional derivative; linearized Crank-Nicolson scheme; unconditional stabilityReferences:
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