  In this work, closure of the Boltzmann--BGK moment hierarchy is accomplished
via projection of the distribution function $f$ onto a space $\mathbb{H}^{N}$
spanned by $N$-order Hermite polynomials. While successive order approximations
retain an increasing number of leading-order moments of $f$, the presented
procedure produces a hierarchy of (single) $N$-order partial-differential
equations providing exact analytical description of the hydrodynamics rendered
by ($N$-order) lattice Boltzmann--BGK (LBGK) simulation. Numerical analysis is
performed with LBGK models and direct simulation Monte Carlo (DSMC) for the
case of a sinusoidal shear wave (Kolmogorov flow) in a wide range of
Weissenberg number $Wi=\tau\nu k^2$ (i.e. Knudsen number $Kn=\lambda
k=\sqrt{Wi}$); $k$ is the wavenumber, $\tau$ the relaxation time of the system,
$\lambda\simeq\tau c_s$ the mean-free path, and $c_s$ the speed of sound. The
present results elucidate the applicability of LBGK simulation under general
non-equilibrium conditions.
