  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.
