  We present a new method to propagate rotating Bose-Einstein condensates
subject to explicitly time-dependent trapping potentials. Using algebraic
techniques, we combine Magnus expansions and splitting methods to yield any
order methods for the multivariate and nonautonomous quadratic part of the
Hamiltonian that can be computed using only Fourier transforms at the cost of
solving a small system of polynomial equations. The resulting scheme solves the
challenging component of the (nonlinear) Hamiltonian and can be combined with
optimized splitting methods to yield efficient algorithms for rotating
Bose-Einstein condensates. The method is particularly efficient for potentials
that can be regarded as perturbed rotating and trapped condensates, e.g., for
small nonlinearities, since it retains the near-integrable structure of the
problem. For large nonlinearities, the method remains highly efficient if
higher order p > 2 is sought. Furthermore, we show how it can adapted to the
presence of dissipation terms. Numerical examples illustrate the performance of
the scheme.
