  The three-dimensional Couette flow between parallel plates is addressed using
mixed lattice Boltzmann models which implement the half-range and the
full-range Gauss-Hermite quadratures on the Cartesian axes perpendicular and
parallel to the walls, respectively. The ability of our models to simulate
rarefied flows are validated through comparison against previously reported
results obtained using the linearized Boltzmann-BGK equation for values of the
Knudsen number (Kn) up to $100$. We find that recovering the non-linear part of
the velocity profile (i.e., its deviation from a linear function) at ${\rm Kn}
\gtrsim 1$ requires high quadrature orders. We then employ the Shakhov model
for the collision term to obtain macroscopic profiles for Maxwell molecules
using the standard $\mu \sim T^\omega$ law, as well as for monatomic Helium and
Argon gases, modeled through ab-initio potentials, where the viscosity is
recovered using the Sutherland model. We validate our implementation by
comparison with DSMC results and find excellent match for all macroscopic
quantities for ${\rm Kn} \lesssim 0.1$. At ${\rm Kn} \gtrsim 0.1$, small
deviations can be seen in the profiles of the diagonal components of the
pressure tensor, the heat flux parallel to the plates, and the velocity
profile, as well as in the values of the velocity gradient at the channel
center. We attribute these deviations to the limited applicability of the
Shakhov collision model for highly out of equilibrium flows.
