  We use strong-coupling perturbation theory, the variational cluster approach
(VCA), and the dynamical density-matrix renormalization group (DDMRG) method to
investigate static and dynamical properties of the one-dimensional
Bose--Hubbard model in both the Mott-insulating and superfluid phases. From the
von Neumann entanglement entropy we determine the central charge and the
transition points for the first two Mott lobes. Our DMRG results for the
ground-state energy, momentum distribution function, boson correlation function
decay, Mott gap, and single particle-spectral function are reproduced very well
by the strong-coupling expansion to fifth order, and by VCA with clusters up to
12 sites as long as the ratio between the hopping amplitude and on-site
repulsion, t/U, is smaller than 0.15 and 0.25, respectively. In addition, in
the superfluid phase VCA captures well the ground-state energy and the sound
velocity of the linear phonon modes. This comparison provides an authoritative
estimate for the range of applicability of these methods. In strong-coupling
theory for the Mott phase, the dynamical structure factor is obtained from the
solution of an effective single-particle problem with an attractive potential.
The resulting resonances show up as double-peak structure close to the
Brillouin zone boundary. These high-energy features also appear in the
superfluid phase which is characterized by a pronounced phonon mode at small
momenta and energies, as predicted by Bogoliubov and field theory. In one
dimension, there are no traces of an amplitude mode in the dynamical
single-particle and two-particle correlation functions.
