  We analyze a large number of high-order discrete velocity models for solving
the Boltzmann-BGK equation for finite Knudsen number flows. Using the
Chapman-Enskog formalism, we prove for isothermal flows a relation identifying
the resolved flow regimes for low Mach numbers. Although high-order lattice
Boltzmann models recover flow regimes beyond the Navier-Stokes level we observe
for several models significant deviations from reference results. We found this
to be caused by their inability to recover the Maxwell boundary condition
exactly. By using supplementary conditions for the gas-surface interaction it
is shown how to systematically generate discrete velocity models of any order
with the inherent ability to fulfill the diffuse Maxwell boundary condition
accurately. Both high-order quadratures and an exact representation of the
boundary condition turn out to be crucial for achieving reliable results. For
Poiseuille flow, we can reproduce the mass flow and slip velocity up to the
Knudsen number of 1. Moreover, for small Knudsen numbers, the Knudsen layer
behavior is recovered.
