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Keywords
(12)
Bose Einstein Condensate
Excited States
Finite Difference
Gradient Flow
Ground State Solution
Monotone Scheme
nonlinear schrodinger equation
Numerical Method
Spectral Method
Crank Nicolson
gross pitaevskii
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Computing the Ground State Solution of BoseEinstein Condensates by a Normalized Gradient Flow
Computing the Ground State Solution of BoseEinstein Condensates by a Normalized Gradient Flow,10.1137/S1064827503422956,Siam Journal on Scientific C
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Computing the Ground State Solution of BoseEinstein Condensates by a Normalized Gradient Flow
(
Citations: 65
)
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Weizhu Bao
,
Qiang Du
In this paper, we prove the energy diminishing of a normalized
gradient flow
which provides a mathematical justification of the imaginary time method used in physical literatures to compute the
ground state solution
of BoseEinstein condensates (BEC). We also investigate the energy diminishing property for the discretization of the normalized gradient flow. Two numerical methods are proposed for such discretizations: one is the backward Euler centered
finite difference
(BEFD), the other one is an explicit timesplitting sinespectral (TSSP) method. Energy diminishing for BEFD and TSSP for linear case, and monotonicity for BEFD for both linear and nonlinear cases are proven. Comparison between the two methods and existing methods, e.g. CrankNicolson
finite difference
(CNFD) or forward Euler
finite difference
(FEFD), shows that BEFD and TSSP are much better in terms of preserving energy diminishing property of the normalized gradient flow. Numerical results in 1d, 2d and 3d with magnetic trap confinement potential, as well as a potential of a stirrer corresponding to a farblue detuned Gaussian laser beam are reported to demonstrate the effectiveness of BEFD and TSSP methods. Furthermore we observe that the normalized
gradient flow
can also be applied directly to compute the first excited state solution in BEC when the initial data is chosen as an odd function.
Journal:
Siam Journal on Scientific Computing
, vol. 25, no. 5, 2004
DOI:
10.1137/S1064827503422956
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Citation Context
(29)
...flow[
4
],andwefollowanartificialrelaxationdynamicsuntilthesystemreachestheequilibriumconfiguration. The gradient flow is introduced by setting φ = φ(τ,x) ,w hereτ is a fictitious time, and solving...
Livio Fedeli
,
et al.
Metastable equilibria of capillary drops on solid surfaces: a phase fi...
...To compute ground states of BECs, Bao and Du [
5
] presented a continuous normalized gradient flow with diminishing energy and discretized it by a backward Euler finite difference method; Bao and Tang [8] proposed a method which can be used to compute the ground and excited states via directly minimizing the energy functional; Edwards and Burnett [19] introduced a Runge–Kuttatype method; other methods include an explicit imaginarytime ...
...The initial data ψ0(z) is chosen as the ground state of the 1D GPE (2.5) with d =1 , γz = 1, and β1 = 50 [
5
, 8]. This corresponds to an experimental setup where initially the condensate is assumed to be in its ground state, and the trap frequency is double at t = 0. We solve this problem by using (3.20) with N = 31 and time step k =0 .001...
...The initial data ψ0(r) is chosen as the ground state of the 2D GPE (2.5) with d =2 ,γr = γx = γy = 1, and β2 = 50 [
5
, 8]. Again this corresponds to an experimental setup where initially the condensate is assumed to be in its ground state, and the trap frequency is doubled at t = 0. We solve this problem by using the TSLP4 method with M = 30 and time step k =0 .001...
...The initial data ψ0(r, z) is chosen as the ground state of the 3D GPE (2.5) with d =3 ,γr = γx = γy =1 ,γz =4 , and β3 = 100 [
5
, 8]. This corresponds to an experimental setup where initially the condensate is assumed to be in its ground state, and at t = 0 we increase the radial frequency four times and decrease the axial frequency to its quarter...
WEIZHU BAO
,
et al.
A FOURTHORDER TIMESPLITTING LAGUERREHERMITE PSEUDOSPECTRAL METHOD F...
...For instance, the GrossPitaevskii equation (GPE) describing BoseEinstein condensates (BEC) [
7
, 23, 29, 54], the Schr˜odingerNewton equation for quantum state reduction [30, 39, 43], the ThomasFermivon Weizs˜acker (TFW), and WangTeter (WT) type orbitalfree model used for electronic structure calculations [16, 19, 33, 51, 52, 53]...
...We should mention that there exist other numerical experiments in the literature, for instance, that of GPE for BoseEinstein condensates [2,
7
, 8] and Schr˜odingerNewton equations for quantum state reduction [30, 39], which coincide with our theory, too...
Huajie Chen
,
et al.
Numerical approximations of a nonlinear eigenvalue problem and applica...
...Most of the numerical algorithms proposed in the literature use the socalled normalized gradient flow [
7
], that consists in applying the steepest descent method for the unconstrained problem,...
...The gradient flow equation (1.4) (or the related continuous gradient flow equation, see [
7
]) can be viewed as a complex heat equation and, consequently, solved by different classical time integration schemes (RungeKuttaFehlberg [15], backward Euler [7, 5, 8], secondorder Strang timesplitting [7, 5], combined RungeKuttaCrankNicolson scheme [3, 4, 11], etc.), and different spatial discretization methods (Fourier spectral [15], finite ...
...The gradient flow equation (1.4) (or the related continuous gradient flow equation, see [7]) can be viewed as a complex heat equation and, consequently, solved by different classical time integration schemes (RungeKuttaFehlberg [15], backward Euler [
7
, 5, 8], secondorder Strang timesplitting [7, 5], combined RungeKuttaCrankNicolson scheme [3, 4, 11], etc.), and different spatial discretization methods (Fourier spectral [15], finite ...
...The gradient flow equation (1.4) (or the related continuous gradient flow equation, see [7]) can be viewed as a complex heat equation and, consequently, solved by different classical time integration schemes (RungeKuttaFehlberg [15], backward Euler [7, 5, 8], secondorder Strang timesplitting [
7
, 5], combined RungeKuttaCrankNicolson scheme [3, 4, 11], etc.), and different spatial discretization methods (Fourier spectral [15], finite ...
... the related continuous gradient flow equation, see [7]) can be viewed as a complex heat equation and, consequently, solved by different classical time integration schemes (RungeKuttaFehlberg [15], backward Euler [7, 5, 8], secondorder Strang timesplitting [7, 5], combined RungeKuttaCrankNicolson scheme [3, 4, 11], etc.), and different spatial discretization methods (Fourier spectral [15], finite elements [5], finite differences [
7
, ...
... flow equation, see [7]) can be viewed as a complex heat equation and, consequently, solved by different classical time integration schemes (RungeKuttaFehlberg [15], backward Euler [7, 5, 8], secondorder Strang timesplitting [7, 5], combined RungeKuttaCrankNicolson scheme [3, 4, 11], etc.), and different spatial discretization methods (Fourier spectral [15], finite elements [5], finite differences [7, 3, 4, 11], sinespectral [
7
], ...
...Since most of the studies [5, 19, 20, 3, 4, 11,
7
, 9] use the imaginary time propagation method (equivalent to the gradient flow model (1.4)), there are few studies using direct minimization by Sobolev gradient methods...
Ionut Danaila
,
et al.
A New Sobolev Gradient Method for Direct Minimization of the GrossPi...
...In [
7
], the ground state solution of Bose–Einstein condensates are determined via a normalized gradient flow discretized by several time integration schemes...
Mark Embree
,
et al.
Dynamical Systems and NonHermitian Iterative Eigensolvers
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, vol. 41, no. 4, pp. 14061426, 2003
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BicelleBound SolidState NMR Structural Studies and MembranePermeabilizing Activities of Piscidin 1 and Piscidin 3: Implications for Mode of Antimicrobial Action
Matthew K. Baxter
,
Jason A. McGavin
,
Nina B. Kraus
,
Anna A. De Angelis
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Jolita Seckute
,
Caitlin Burzynski
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Daryl M. Berke
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Nedzada Smajic
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Linda K. Nicholson
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Myriam Cotten
Journal:
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, vol. 100, no. 3, pp. 496a496a, 2011
Exploring bistability in rotating Bose–Einstein condensates by a quotient transformation invariant continuation method
YuehCheng Kuo
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WenWei Lin
,
ShihFeng Shieh
,
Weichung Wang
Journal:
Physica Dnonlinear Phenomena  PHYSICA D
, vol. 240, no. 1, pp. 7888, 2011
Metastable equilibria of capillary drops on solid surfaces: a phase field approach
Livio Fedeli
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Alessandro Turco
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Antonio Desimone
Journal:
Continuum Mechanics and Thermodynamics  CONTINUUM MECH THERMODYN
, vol. 23, pp. 119, 2011
A spectralGalerkin continuation method using Chebyshev polynomials for the numerical solutions of the GrossPitaevskii equation
Y.S. Wang
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C.S. Chien
Journal:
Journal of Computational and Applied Mathematics  J COMPUT APPL MATH
, vol. 235, no. 8, pp. 27402757, 2011
A FOURTHORDER TIMESPLITTING LAGUERREHERMITE PSEUDOSPECTRAL METHOD FOR BOSEEINSTEIN CONDENSATES
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Citations: 17
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WEIZHU BAO
,
JIE SHEN
Journal:
Siam Journal on Scientific Computing
, 2010