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Mathematics of Computation
Mathematics of Computation
ISSN 1088-6842(online) ISSN 0025-5718(print)

 

Optimal error estimates of finite difference methods for the Gross-Pitaevskii equation with angular momentum rotation


Authors: Weizhu Bao and Yongyong Cai
Journal: Math. Comp. 82 (2013), 99-128
MSC (2010): Primary 35Q55, 65M06, 65M12, 65M22, 81-08
Published electronically: June 20, 2012
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Abstract: We analyze finite difference methods for the Gross-Pitaevskii equation with an angular momentum rotation term in two and three dimensions and obtain the optimal convergence rate, for the conservative Crank-Nicolson finite difference (CNFD) method and semi-implicit finite difference (SIFD) method, at the order of $ O(h^2+\tau ^2)$ in the $ l^2$-norm and discrete $ H^1$-norm with time step $ \tau $ and mesh size $ h$. Besides the standard techniques of the energy method, the key technique in the analysis for the SIFD method is to use the mathematical induction, and resp., for the CNFD method is to obtain a priori bound of the numerical solution in the $ l^\infty $-norm by using the inverse inequality and the $ l^2$-norm error estimate. In addition, for the SIFD method, we also derive error bounds on the errors between the mass and energy in the discretized level and their corresponding continuous counterparts, respectively, which are at the same order of the convergence rate as that of the numerical solution itself. Finally, numerical results are reported to confirm our error estimates of the numerical methods.


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Additional Information

Weizhu Bao
Affiliation: Department of Mathematics and Center for Computational Science and Engineering, National University of Singapore, Singapore 119076
Email: bao@math.nus.edu.sg

Yongyong Cai
Affiliation: Department of Mathematics, National University of Singapore, Singapore 119076
Email: caiyongyong@nus.edu.sg

DOI: http://dx.doi.org/10.1090/S0025-5718-2012-02617-2
PII: S 0025-5718(2012)02617-2
Keywords: Gross-Pitaevskii equation, angular momentum rotation, finite difference method, semi-implicit scheme, conservative Crank-Nicolson scheme
Received by editor(s): March 29, 2010
Received by editor(s) in revised form: March 26, 2011
Published electronically: June 20, 2012
Article copyright: © Copyright 2012 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.