  In this paper, we propose an efficient and accurate numerical method for
computing the ground state of spin-1 Bose-Einstein condensates (BEC) by using
the normalized gradient flow or imaginary time method. The key idea is to find
a third projection or normalization condition based on the relation between the
chemical potentials so that the three projection parameters used in the
projection step of the normalized gradient flow are uniquely determined by this
condition as well as the other two physical conditions given by the
conservation of total mass and total magnetization. This allows us to
successfully extend the most popular and powerful normalized gradient flow or
imaginary time method for computing the ground state of single component BEC to
compute the ground state of spin-1 BEC. An efficient and accurate
discretization scheme, the backward-forward Euler sine-pseudospectral method
(BFSP), is proposed to discretize the normalized gradient flow. Extensive
numerical results on ground states of spin-1 BEC with
ferromagnetic/antiferromagnetic interaction and harmonic/optical lattice
potential in one/three dimensions are reported to demonstrate the efficiency of
our new numerical method.
