  We analyze the time evolution of the Bose-Hubbard model after a sudden
quantum quench to a weakly interacting regime. Specifically, motivated by a
recent experiment at Kyoto University, we numerically simulate redistribution
of the kinetic and onsite-interaction energies at an early time, which was
observed in non-equilibrium dynamics of ultracold Bose gases in a cubic optical
lattice starting with a singly-occupied Mott-insulator state. In order to
compute the short-time dynamics corresponding to the experimental situation, we
apply the truncated-Wigner approximation (TWA) to the Bose-Hubbard model on a
cubic lattice. We show that our semiclassical approach quantitatively
reproduces the fast redistribution dynamics. We further analyze spatial
spreading of density-density correlations at equal time in the Bose-Hubbard
model on a square lattice with a large filling factor. When the system is
initially prepared in a coherent state, we find that a propagation velocity of
the correlation wave packet in the correlation function strongly depends on the
final interaction strength, and it is bounded by twice the maximum group
velocity of the elementary excitations. In contrast, when the system is
initially in a Mott-insulator state, the propagation velocity of the wave
packet is approximately independent of the final interaction strength.
