  Second-order flows in this paper refer to some artificial evolutionary
differential equations involving second-order time derivatives distinguished
from gradient flows which are considered to be first-order flows. This is a
popular topic due to the recent advances of inertial dynamics with damping in
convex optimization. Mathematically, the ground state of a rotating
Bose-Einstein condensate (BEC) can be modeled as a minimizer of the
Gross-Pitaevskii energy functional with angular momentum rotational term under
the normalization constraint. We introduce two types of second-order flows as
energy minimization strategies for this constrained non-convex optimization
problem, in order to approach the ground state. The proposed artificial
dynamics are novel second-order nonlinear hyperbolic partial differential
equations with dissipation. Several numerical discretization schemes are
discussed, including explicit and semi-implicit methods for temporal
discretization, combined with a Fourier pseudospectral method for spatial
discretization. These provide us a series of efficient and robust algorithms
for computing the ground states of rotating BECs. Particularly, the newly
developed algorithms turn out to be superior to the state-of-the-art numerical
methods based on the gradient flow. In comparison with the gradient flow type
approaches: When explicit temporal discretization strategies are adopted, the
proposed methods allow for larger stable time step sizes; While for
semi-implicit discretization, using the same step size, a much smaller number
of iterations are needed for the proposed methods to reach the stopping
criterion, and every time step encounters almost the same computational
complexity. Rich and detailed numerical examples are documented for
verification and comparison.
