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A NEW LINEAR AND CONSERVATIVE FINITE DIFFERENCE SCHEME FOR THE GROSS–PITAEVSKII EQUATION WITH ANGULAR MOMENTUM ROTATION

Part of: Partial differential equations, initial value and time-dependent initial-boundary value problems

Published online by Cambridge University Press:  08 April 2019

JIN CUI Open the ORCID record for JIN CUI [Opens in a new window] ,
WENJUN CAI ,
CHAOLONG JIANG  and
YUSHUN WANG
JIN CUI
Affiliation:
School of Mathematical Sciences, Jiangsu Key Laboratory for NSLSCS, Nanjing Normal University, Nanjing 210023, China email caiwenjun@njnu.edu.cn, wangyushun@njnu.edu.cn Department of Basic Sciences, Nanjing Vocational College of Information Technology, Nanjing 210023, China email seucj008@163.com
WENJUN CAI
Affiliation:
School of Mathematical Sciences, Jiangsu Key Laboratory for NSLSCS, Nanjing Normal University, Nanjing 210023, China email caiwenjun@njnu.edu.cn, wangyushun@njnu.edu.cn
CHAOLONG JIANG
Affiliation:
School of Statistics and Mathematics, Yunnan University of Finance and Economics, Kunming 650221, China email Chaolong_jiang@126.com
YUSHUN WANG*
Affiliation:
School of Mathematical Sciences, Jiangsu Key Laboratory for NSLSCS, Nanjing Normal University, Nanjing 210023, China email caiwenjun@njnu.edu.cn, wangyushun@njnu.edu.cn
*
✉wangyushun@njnu.edu.cn
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Abstract

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A new linear and conservative finite difference scheme which preserves discrete mass and energy is developed for the two-dimensional Gross–Pitaevskii equation with angular momentum rotation. In addition to the energy estimate method and mathematical induction, we use the lifting technique as well as some well-known inequalities to establish the optimal $H^{1}$-error estimate for the proposed scheme with no restrictions on the grid ratio. Unlike the existing numerical solutions which are of second-order accuracy at the most, the convergence rate of the numerical solution is proved to be of order $O(h^{4}+\unicode[STIX]{x1D70F}^{2})$ with time step $\unicode[STIX]{x1D70F}$ and mesh size $h$. Numerical experiments have been carried out to show the efficiency and accuracy of our new method.

Keywords

Gross–Pitaevskii equationangular momentum rotationfinite difference schemeconservation lawerror estimate

MSC classification

Primary: 65M06: Finite difference methods
Secondary: 65M12: Stability and convergence of numerical methods
Type
Research Article
Information
The ANZIAM Journal , Volume 61 , Issue 2 , April 2019 , pp. 204 - 232
Copyright
© 2019 Australian Mathematical Society 

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