  We use tensor network methods - Matrix Product States, Tree Tensor Networks,
and Locally Purified Tensor Networks - to simulate the one dimensional
Bose-Hubbard model for zero and finite temperatures in experimentally
accessible regimes. We first explore the effect of thermal fluctuations on the
system ground state by characterizing its Mott and superfluid features. Then,
we study the behavior of the out-of-equilibrium dynamics induced by quenches of
the hopping parameter. We confirm a Kibble-Zurek scaling for zero temperature
and characterize the finite temperature behavior, which we explain by means of
a simple argument.
