  We investigate the equilibrium and dynamical properties of the Bose-Hubbard
model and the related particle-hole symmetric spin-1 model in the vicinity of
the superfluid to Mott insulator quantum phase transition. We employ the
following methods: exact-diagonalization, mean field (Gutzwiller), cluster
mean-field, and mean-field plus Gaussian fluctuations. In the first part of the
paper we benchmark the four methods by analyzing the equilibrium problem and
give numerical estimates for observables such as the density of double
occupancies and their correlation function. In the second part, we study
parametric ramps from the superfluid to the Mott insulator and map out the
crossover from the regime of fast ramps, which is dominated by local physics,
to the regime of slow ramps with a characteristic universal power law scaling,
which is dominated by long wavelength excitations. We calculate values of
several relevant physical observables, characteristic time scales, and an
optimal protocol needed for observing universal scaling.
