  We use Lattice-Boltzmann and analytical calculations to investigate transient
hydrodynamic finite size effects induced by the use of periodic boundary
conditions in simulations at the molecular, mesoscopic or continuum levels of
description. We analyze the transient response to a local perturbation in the
fluid and obtain via linear response theory the local velocity correlation
function. This new approach is validated by comparing the finite size effects
on the steady-state velocity with the known results for the diffusion
coefficient. We next investigate the full time-dependence of the local velocity
auto-correlation function. We find at long times a cross-over between the
expected $t^{-3/2}$ hydrodynamic tail and an oscillatory exponential decay, and
study the scaling with the system size of the cross-over time, exponential rate
and amplitude, and oscillation frequency. We interpret these results from the
analytic solution of the compressible Navier-Stokes equation for the slowest
modes, which are set by the system size. The present work not only provides a
comprehensive analysis of hydrodynamic finite size effects in bulk fluids, but
also establishes the Lattice-Boltzmann method as a suitable tool to investigate
such effects in general.
