A new high-accuracy difference method for nonhomogeneous time-fractional Schrödinger equation

Abstract

The fractional Schrödinger equation is an important fractional nonlinear evolution equation, and the study of its numerical solution has profound scientific meaning and wide application prospects. This paper proposes a new high-accuracy difference method for nonhomogeneous time-fractional Schrödinger equation (TFSE). The Caputo time-fractional derivative is discretized by high-order $L2-1_{\sigma }$ formula and the fourth-order compact difference approximation is applied for spatial discretization. A new nonlinear compact difference scheme with temporal second-order and spatial fourth-order accuracy is constructed, which is solved by the efficient linearized iterative algorithm. The unconditional stability and convergence are analysed by the energy method. The unique existence and maximum-norm estimate of new compact difference scheme solution are obtained. Theoretical analysis shows that the convergence accuracy of new compact difference scheme is $O({\tau }^2+h^4)$ with the strong regularity assumption. Numerical experiments verify theoretical results and indicate that the proposed method is an efficient numerical method for solving TFSE.

Keywords:

• Nonhomogeneous time-fractional Schrödinger equation (TFSE)
• compact difference scheme
• unconditional stability
• convergence order
• numerical experiments

2020 Mathematics Subject Classifications:

• 65M06
• 65M12
• 65M15

Acknowledgments

The authors express the sincere gratitude to Dr. Lifei Wu and Dr. Xinlong Liu (North China Electric Power University) for their effective advice.

Disclosure statement

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

Additional information

Funding

This research is supported by the National Natural Science Foundation of China (Grant No. 11371135).