Efficient kinetic method for fluid simulation beyond the Navier-Stokes   equation
  We present a further theoretical extension to the kinetic theory based formulation of the lattice Boltzmann method of Shan et al (2006). In addition to the higher order projection of the equilibrium distribution function and a sufficiently accurate Gauss-Hermite quadrature in the original formulation, a new regularization procedure is introduced in this paper. This procedure ensures a consistent order of accuracy control over the non-equilibrium contributions in the Galerkin sense. Using this formulation, we construct a specific lattice Boltzmann model that accurately incorporates up to the third order hydrodynamic moments. Numerical evidences demonstrate that the extended model overcomes some major defects existed in the conventionally known lattice Boltzmann models, so that fluid flows at finite Knudsen number (Kn) can be more quantitatively simulated. Results from force-driven Poiseuille flow simulations predict the Knudsen's minimum and the asymptotic behavior of flow flux at large Kn. 
Raoyang Zhang, Xiaowen Shan and Hudong Chen