Some recent results on discrete velocity models and ramifications for lattice Boltzmann equation

Some rigorous results on discrete velocity models are briefly reviewed and their ramifications for the lattice Boltzmann equation (LBE) are discussed. In particular, issues related to thermodynamics and H-theorem of the latti ce Boltzmann equation are addressed. It is argued that for the lattice Bolt zmann equation satisfying the correct hydrodynamic equations, there cannot exist an H-theorem. Nevertheless, the equilibrium distribution function of the lattice Boltzmann equation can closely approximate the genuine equilibr ium which minimizes the H-function of the corresponding continuous Boltzman n equation. It is also pointed out that the "equilibrium" in the LEE models is an attractor rather than a true equilibrium in the rigorous sense of H- theorem. Since there is no H-theorem to guarantee the stability of the LEE models at the attractor, the stability of the attractor can only be studied by means other than proving an H-function. (C) 2000 Elsevier Science B.V. All rights reserved.