High-order numerical methods with mass and energy conservation for spin–orbit-coupled Bose–Einstein condensates

ABSTRACT

In this work, a high-order numerical method with mass and energy conservation obtained by the operator-compensation (OC) technique is proposed for the coupled Gross–Pitaevskii equations (CGPEs) modelling the spin–orbit-coupled Bose–Einstein condensates. The OC technique is utilized to seek high-order approximations to the first- and the second-order spatial derivatives in the CGPEs, after which a time-dependent semi-discrete system is obtained. Then we obtain a fully discrete scheme by applying the Crank–Nicolson method to discretize this system in time. We theoretically show that the semi-discrete system and the fully discrete scheme conserve both mass and energy on the discrete level. In addition, the stability analysis is discussed for the proposed method. Numerical experiments are given to test accuracy and verify conservative properties. The dynamics of the two-dimensional CGPEs and the effects of the parameters in the CGPEs are simulated.

KEYWORDS:

• Spin–orbit-coupled Bose–Einstein condensates
• coupled Gross–Pitaevskii equations
• operator-compensation method
• high-order conservative method
• dynamics

2010 MATHEMATICS SUBJECT CLASSIFICATIONS:

• 65M06
• 35L05
• 81-08

Disclosure statement

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

Additional information

Funding

This work was supported by the National Natural Science Foundation of China (Project No. 11671044), the Science Challenge Project (Project No. TZ2016001), National Key Laboratory of Computational Physics (Project No. 6142A05180103) and Hunan Provincial Innovation Foundation for Postgraduate (Project No. CX2018B380).