  In this paper, we propose an efficient and spectrally accurate numerical
method for computing the dynamics of rotating Bose-Einstein condensates (BEC)
in two dimensions (2D) and 3D based on the Gross-Pitaevskii equation (GPE) with
an angular momentum rotation term. By applying a time-splitting technique for
decoupling the nonlinearity and properly using the alternating direction
implicit (ADI) technique for the coupling in the angular momentum rotation term
in the GPE, at every time step, the GPE in rotational frame is decoupled into a
nonlinear ordinary differential equation (ODE) and two partial differential
equations with constant coefficients. This allows us to develop new
time-splitting spectral (TSSP) methods for computing the dynamics of BEC in a
rotational frame.The new numerical method is explicit, unconditionally stable,
and of spectral accuracy in space and second order accuracy in time. Moreover,
it is time reversible and time transverse invariant, and conserves the position
density in the discretized level if the GPE does. Extensive numerical results
are presented to confirm the above properties of the new numerical method for
rotating BEC in 2D and 3D.
