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 andH -theorem of the lattice Boltzmann equation are addressed. It is argued that for the lattice Boltzmann 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 equilibrium which minimizes the H -function of the corresponding continuous Boltzmann equation. It is also pointed out that the “equilibrium” in the LBE 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 LBE models at the attractor, the stability of the attractor can only be studied by means other than proving an H -function.  2000 Elsevier Science B.V. All rights reserved.