A fast implicit difference scheme for solving high-dimensional time-space fractional nonlinear Schrödinger equation

Abstract

In this work, an efficient implicit difference scheme is developed for solving the high-dimensional time-space fractional nonlinear Schrödinger equation. The derived scheme is constructed by utilizing a fast evaluation of Caputo fractional derivative based on the $\mathcal{L}2$-$1_{\sigma }$ formula; meanwhile, the compact finite difference with matrix transfer technique is adopted for the spatial discretization. Moreover, a linearized iteration method based on the fast discrete sine transform technique is considered to solve the nonlinear system effectively. Because the resultant algorithm does not use matrix inversion, it is computationally efficient in long-time calculations. The stability, truncation error and convergence analysis of the discrete scheme are discussed in detail. Furthermore, a fast iterative algorithm is provided. Finally, several numerical examples are given to verify the efficiency and accuracy of the derived scheme, and a comparison with similar work is presented.

Keywords:

• Time-space fractional Schrödinger equation
• matrix transfer technique
• fast algorithm
• stability

AMS SUBJECT CLASSIFICATIONS:

• 35J10
• 65N06
• 65N12
• 65N22
• 65Y20

Acknowledgements

The author is very grateful to the referees for their constructive comments and valuable suggestions, which lead to an improvement of this paper.

Disclosure statement

No potential conflict of interest was reported by the author(s).