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We propose a preconditioned nonlinear conjugate gradient method coupled witha spectral spatial dis-cretization scheme for computing the ground states (GS)of rotating Bose-Einstein condensates (BEC), modeled by the Gross-PitaevskiiEquation (GPE). We first start by reviewing the classical gradient flow (alsoknown as imaginary time (IMT)) method which considers the problem from the PDEstandpoint, leading to numerically solve a dissipative equation. Based on thisIMT equation, we analyze the forward Euler (FE), Crank-Nicolson (CN) and theclassical backward Euler (BE) schemes for linear problems and recognizeclassical power iterations, allowing us to derive convergence rates. Byconsidering the alternative point of view of minimization problems, we proposethe preconditioned gradient (PG) and conjugate gradient (PCG) methods for theGS computation of the GPE. We investigate the choice of the preconditioner,which plays a key role in the acceleration of the convergence process. Theperformance of the new algorithms is tested in 1D, 2D and 3D. We conclude thatthe PCG method outperforms all the previous methods, most particularly for 2Dand 3D fast rotating BECs, while being simple to implement.

journal of computational physics

Efficient spectral computation of the stationary states of rotating  Bose-Einstein condensates by the preconditioned nonlinear conjugate gradient  method