Error analysis of a fully discrete scheme for time fractional Schrödinger equation with initial singularity

Abstract

We consider the numerical approximation for a time fractional Schrödinger equation whose solution exhibits an initial weak singularity. A fully discrete scheme is constructed using $L1$ scheme on graded mesh for the discretiaztion of temporal Caputo derivative and spectral method for spatial discretization. It is shown that with appropriate choice of the grading parameter, the scheme can attain order $2-\alpha$ convergence in temporal direction, where α $(0<\alpha <1)$ is the order of time Caputo fractional derivative, and spectral accuracy in spatial direction if the solution is sufficiently smooth in its spatial part. Numerical results confirm the sharpness of the error analysis.

Keywords:

• Time fractional Schrödinger equation
• spectral method
• weak singularity
• graded mesh
• convergence

2010 Mathematics Subject Classifications:

• 65L05
• 65L11
• 26A33

Acknowledgments

We would like to thank the anonymous referees for their valuable suggestions and comments, which greatly helped improve the presentation of this manuscript.

Disclosure statement

No potential conflict of interest was reported by the authors.

Additional information

Funding

The work of Jun Zhang is supported in part by China Postdoctoral Science Foundation (No. 2019M653490) and the Academic Project of Guizhou University of Finance and Economics (No. [2018]5774-033); the work of Hu Chen is supported in part by NSF of China (No. 11801026) and China Postdoctoral Science Foundation (No. 2018M631316); the work of Tao Sun is supported in part by NSF of China (Nos. 11401380, 11671166 and 11701371); the work of Jinrong Wang is supported in part by NSF of China (Nos. 11661016 and 11461012).