  A novel discretization approach for the Bhatnager-Gross-Krook (BGK) kinetic
equation is proposed. An hierarchy of LB models starting from $D1Q3$ model with
increasing number of velocities converging to BGK model is derived. The method
inherits properties of the Lattice Boltzmann (LB) method like linear streaming
step, conservation of moments. Similar to the finite-difference methods for the
BGK model the presented approach describes high-order moments of the
distribution function with controllable error. The Sod shock tube problem, the
Poiseuille flow between parallel plates and the plane Couette flow are
considered for wide range of Knudsen numbers. Good stability and significant
increase in precision over the conventional LB models are observed.
