Optimal error estimates of explicit finite difference schemes for the coupled Gross–Pitaevskii equations

ABSTRACT

In this paper, we investigate the convergence of explicit finite difference schemes which contain a Richardson scheme and a leap-frog scheme for computing the coupled Gross–Pitaevskii equations in high space dimensions. We establish the optimal error estimates of our schemes at the order of $O({\tau }^2+h^2)$ in the $l^{\mathrm{\infty }}$-norm with the time step τ and the mesh size h. Besides the standard techniques of the energy analysis method, the key techniques in the analysis is to use the method of induction argument and order reduction. The numerical results are reported to confirm our theoretical analysis for the numerical methods.

KEYWORDS:

• Coupled Gross–Pitaevskii equations
• Bose–Einstein condensation
• rotating Lagrangian coordinate
• induction argument method
• convergence

AMS SUBJECT CLASSIFICATIONS:

• 65M06
• 35Q55
• 35Q51

Acknowledgements

The authors are very grateful to both reviewers for carefully reading the paper and for their comments and suggestions which have led to improvement of the paper.

Disclosure statement

No potential conflict of interest was reported by the authors.

ORCID

F. Liao  http://orcid.org/0000-0003-2457-4458

Additional information

Funding

This work is supported by Jiangsu Innovation Program for Graduate Education under Grant No. KYZZ16_0161 and the National Natural Science Foundation of China under Grant No. 11571181.