  We propose a simple, efficient and accurate numerical method for simulating
the dynamics of rotating Bose-Einstein condensates (BECs) in a rotational frame
with/without a long-range dipole-dipole interaction. We begin with the
three-dimensional (3D) Gross-Pitaevskii equation (GPE) with an angular momentum
rotation term and/or long-range dipole-dipole interaction, state the
two-dimensional (2D) GPE obtained from the 3D GPE via dimension reduction under
anisotropic external potential and review some dynamical laws related to the 2D
and 3D GPE. By introducing a rotating Lagrangian coordinate system, the
original GPEs are re-formulated to GPEs without the angular momentum rotation
which is replaced by a time-dependent potential in the new coordinate system.
We then cast the conserved quantities and dynamical laws in the new rotating
Lagrangian coordinates. Based on the new formulation of the GPE for rotating
BECs in the rotating Lagrangian coordinates, a time-splitting spectral method
is presented for computing the dynamics of rotating BECs. The new numerical
method is explicit, simple to implement, unconditionally stable and very
efficient in computation. It is spectral order accurate in space and
second-order accurate in time, and conserves the mass in the discrete level.
Extensive numerical results are reported to demonstrate the efficiency and
accuracy of the new numerical method. Finally, the numerical method is applied
to test the dynamical laws of rotating BECs such as the dynamics of condensate
width, angular momentum expectation and center-of-mass, and to investigate
numerically the dynamics and interaction of quantized vortex lattices in
rotating BECs without/with the long-range dipole-dipole interaction.
