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Accurate and Efficient Numerical Methods for Computing Ground States and Dynamics of Dipolar Bose-Einstein Condensates via the Nonuniform FFT

Part of: Numerical methods in Fourier analysis Quantum theory Equations of mathematical physics and other areas of application Partial differential equations, initial value and time-dependent initial-boundary value problems

Published online by Cambridge University Press:  17 May 2016

Weizhu Bao ,
Qinglin Tang  and
Yong Zhang
Weizhu Bao*
Affiliation:
Department of Mathematics, National University of Singapore, Singapore 119076
Qinglin Tang*
Affiliation:
Université de Lorraine, Institut Elie Cartan de Lorraine, UMR 7502 , Vandoeuvre-lès-Nancy, F-54506, France Inria Nancy Grand-Est/IECL-CORIDA, France Beijing Computational Science Research Center, No. 10 West Dongbeiwang Road, Beijing 100094, P.R. China
Yong Zhang*
Affiliation:
Université de Rennes 1, IRMAR, Campus de Beaulieu, 35042 Rennes Cedex, France Wolfgang Pauli Institute c/o Fak. Mathematik, University Wien, Oskar-Morgenstern-Platz 1, 1090 Vienna, Austria
*
*Corresponding author. Email addresses:matbaowz@nus.edu.sg(W. Bao), tqltql2010@gmail.com (Q. Tang), yong.zhang@univ-rennes1.fr(Y. Zhang)
*Corresponding author. Email addresses:matbaowz@nus.edu.sg(W. Bao), tqltql2010@gmail.com (Q. Tang), yong.zhang@univ-rennes1.fr(Y. Zhang)
*Corresponding author. Email addresses:matbaowz@nus.edu.sg(W. Bao), tqltql2010@gmail.com (Q. Tang), yong.zhang@univ-rennes1.fr(Y. Zhang)
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Abstract

We propose efficient and accurate numerical methods for computing the ground state and dynamics of the dipolar Bose-Einstein condensates utilising a newly developed dipole-dipole interaction (DDI) solver that is implemented with the non-uniform fast Fourier transform (NUFFT) algorithm. We begin with the three-dimensional (3D) Gross-Pitaevskii equation (GPE) with a DDI term and present the corresponding two-dimensional (2D) model under a strongly anisotropic confining potential. Different from existing methods, the NUFFT based DDI solver removes the singularity by adopting the spherical/polar coordinates in the Fourier space in 3D/2D, respectively, thus it can achieve spectral accuracy in space and simultaneously maintain high efficiency by making full use of FFT and NUFFT whenever it is necessary and/or needed. Then, we incorporate this solver into existing successful methods for computing the ground state and dynamics of GPE with a DDI for dipolar BEC. Extensive numerical comparisons with existing methods are carried out for computing the DDI, ground states and dynamics of the dipolar BEC. Numerical results show that our new methods outperform existing methods in terms of both accuracy and efficiency.

Keywords

Dipolar BECdipole-dipole interactionNUFFTground statedynamicscollapse

MSC classification

Secondary: 35Q40: PDEs in connection with quantum mechanics 35Q41: Time-dependent Schrödinger equations, Dirac equations 35Q55: NLS-like equations (nonlinear Schrödinger) 65M70: Spectral, collocation and related methods 65T40: Trigonometric approximation and interpolation 65T50: Discrete and fast Fourier transforms 81-08: Computational methods
Type
Research Article
Information
Communications in Computational Physics , Volume 19 , Issue 5 , May 2016 , pp. 1141 - 1166
Copyright
Copyright © Global-Science Press 2016 

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