  In this paper, we prove that a local weak solution to the $d$-dimensional
incompressible Navier-Stokes equations ($d \geq 2$) can be constructed by
taking the hydrodynamic limit of a velocity-discretized Boltzmann equation with
a simplified BGK collision operator. Moreover, in the case when the dimension
is $d=2,3$, we characterize the combinations of finitely many particle
velocities and probabilities that lead to the incompressible Navier-Stokes
equations in the hydrodynamic limit. Numerical computations conducted in 2D
provide information about the rate with which this hydrodynamic limit is
achieved when the Knudsen number tends to zero.
