** Kinetic theory representation of hydrodynamics: a way behond the Navier-Stokes equation, **
*J. Fluid Mech. 550, 413-441*

This paper proposes a systematic approach to deriving a discrete lattice Boltzmann scheme from the continuum Boltzmann equation. Inspired by the Grad-13 moment system, the Boltzmann distribution function is expanded on a Hermite basis in velocity space. Unlike Grad's approach, the paper focuses however not on the moment representation, but on a discrete set of kinetic variables, which are recovered by a reverse calculation from the set of moments. A Gauss-Hermite quadrature is used to integrate the hydrodynamic moments. It is shown that the discrete velocity set of the lattice Boltzmann scheme is identical with the quadrature points of this procedure. The paper extends previous results in He and Luo 97 and He et al. 98 to create a consistent theoretical framework for the lattice Boltzmann method. In particular, the D2Q17 and D3Q39 lattices are introduced, in which thermal effects or phenomena at higher Knudsen number can be modeled thanks to an enlarged neighborhood for inter-cell communication.