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We analyze a large number of high-order discrete velocity models for solvingthe Boltzmann-BGK equation for finite Knudsen number flows. Using theChapman-Enskog formalism, we prove for isothermal flows a relation identifyingthe resolved flow regimes for low Mach numbers. Although high-order latticeBoltzmann models recover flow regimes beyond the Navier-Stokes level we observefor several models significant deviations from reference results. We found thisto be caused by their inability to recover the Maxwell boundary conditionexactly. By using supplementary conditions for the gas-surface interaction itis shown how to systematically generate discrete velocity models of any orderwith the inherent ability to fulfill the diffuse Maxwell boundary conditionaccurately. Both high-order quadratures and an exact representation of theboundary condition turn out to be crucial for achieving reliable results. ForPoiseuille flow, we can reproduce the mass flow and slip velocity up to theKnudsen number of 1. Moreover, for small Knudsen numbers, the Knudsen layerbehavior is recovered.

High-order lattice Boltzmann models for wall-bounded flows at finite Knudsen numbers