Error analysis of a time fourth-order exponential wave integrator Fourier pseudo-spectral method for the nonlinear Dirac equation

Abstract

In this paper, we propose a time fourth-order exponential wave integrator (EWI) Fourier pseudo-spectral method for solving the nonlinear Dirac equation with periodic boundary value conditions. The new numerical method is designed by using Fourier pseudo-spectral method for spatial derivatives combined with a new fourth-order EWI for temporal derivatives. We give rigourously error analysis and establish error bounds for the numerical solutions without any CFL-type condition constraint. In more details, the proposed method has the fourth-order temporal accuracy and spectral spatial accuracy, respectively, in general $H^m$-norm. Extensive numerical experiments are reported to confirm the theoretical analysis and show the superiority of the new method.

Keywords:

• Nonlinear Dirac equation
• exponential wave integrator
• Fourier pseudo-spectral method
• error analysis
• fourth-order method

2010 Mathematics Subject Classifications:

• 65M15
• 65M70

Acknowledgments

The authors are grateful to the anonymous reviewers for their valuable suggestions, which help improve this paper significantly.

Disclosure statement

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

Additional information

Funding

The research was supported in part by Education Department of Hebei Province Fund under Grant Number QN2019053 and Hebei Natural Science Foundation of China under Grant Number A2014205136.