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Computing the Ground State Solution of Bose--Einstein Condensates by a Normalized Gradient Flow

Computing the Ground State Solution of Bose--Einstein Condensates by a Normalized Gradient Flow,10.1137/S1064827503422956,Siam Journal on Scientific C

Computing the Ground State Solution of Bose--Einstein Condensates by a Normalized Gradient Flow   (Citations: 65)
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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 Bose-Einstein 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 time-splitting sine-spectral (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. Crank-Nicolson 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 far-blue 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
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    • ...flow[4],andwefollowanartificialrelaxationdynamicsuntilthesystemreachestheequilibriumconfiguration. The gradient flow is introduced by setting φ = φ(τ,x) ,w hereτ is a fictitious time, and solving...

    Livio Fedeliet 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–Kutta-type method; other methods include an explicit imaginary-time ...
    • ...The initial data ψ0(z) is chosen as the ground state of the 1-D 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 2-D 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 3-D 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 BAOet al. A FOURTH-ORDER TIME-SPLITTING LAGUERRE-HERMITE PSEUDOSPECTRAL METHOD F...

    • ...For instance, the Gross-Pitaevskii equation (GPE) describing Bose-Einstein condensates (BEC) [7, 23, 29, 54], the Schr˜odinger-Newton equation for quantum state reduction [30, 39, 43], the Thomas-Fermi-von Weizs˜acker (TFW), and Wang-Teter (WT) type orbital-free 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 Bose-Einstein condensates [2, 7, 8] and Schr˜odinger-Newton equations for quantum state reduction [30, 39], which coincide with our theory, too...

    Huajie Chenet al. Numerical approximations of a nonlinear eigenvalue problem and applica...

    • ...Most of the numerical algorithms proposed in the literature use the so-called 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 (Runge-Kutta-Fehlberg [15], backward Euler [7, 5, 8], second-order Strang time-splitting [7, 5], combined Runge-Kutta-Crank-Nicolson 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 (Runge-Kutta-Fehlberg [15], backward Euler [7, 5, 8], second-order Strang time-splitting [7, 5], combined Runge-Kutta-Crank-Nicolson 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 (Runge-Kutta-Fehlberg [15], backward Euler [7, 5, 8], second-order Strang time-splitting [7, 5], combined Runge-Kutta-Crank-Nicolson 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 (Runge-Kutta-Fehlberg [15], backward Euler [7, 5, 8], second-order Strang time-splitting [7, 5], combined Runge-Kutta-Crank-Nicolson 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 (Runge-Kutta-Fehlberg [15], backward Euler [7, 5, 8], second-order Strang time-splitting [7, 5], combined Runge-Kutta-Crank-Nicolson scheme [3, 4, 11], etc.), and different spatial discretization methods (Fourier spectral [15], finite elements [5], finite differences [7, 3, 4, 11], sine-spectral [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 Danailaet al. A New Sobolev Gradient Method for Direct Minimization of the Gross--Pi...

    • ...In [7], the ground state solution of Bose–Einstein condensates are determined via a normalized gradient flow discretized by several time integration schemes...

    Mark Embreeet al. Dynamical Systems and Non-Hermitian Iterative Eigensolvers

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