  We study numerically the time-independent vector Gross-Pitaevskii equations
(VGPEs) for ground states and time-dependent VGPEs with (or without) an
external driven field for dynamics describing a multi-component Bose-Einstein
condensate (BEC) at zero or very low temperature. In preparation for the
numerics, we scale the 3d VGPEs, approximately reduce it to lower dimensions,
present a normalized gradient flow (NGF) to compute ground states of
multi-component BEC, prove energy diminishing of the NGF which provides a
mathematical justification, discretize it by the backward Euler finite
difference (BEFD) which is monotone in linear and nonlinear cases and preserves
energy diminishing property in linear case. Then we use a time-splitting
sine-spectral method (TSSP) to discretize the time-dependent VGPEs with an
external driven field for computing dynamics of multi-component BEC. The merit
of the TSSP for VGPEs is that it is explicit, unconditionally stable, time
reversible and time transverse invariant if the VGPEs is, `good' resolution in
the semiclassical regime, of spectral order accuracy in space and second order
accuracy in time, and conserves the total particle number in the discretized
level. Extensive numerical examples in 3d for ground states and dynamics of
multi-component BEC are presented to demonstrate the power of the numerical
methods and to discuss the physics of multi-component Bose-Einstein
condensates.
