  The particles model, the collision model, the polynomial development used for
the equilibrium distribution, the time discretization and the velocity
discretization are factors that let the lattice Boltzmann framework (LBM) far
away from its conceptual support: the continuous Boltzmann equation (BE). Most
collision models are based on the BGK, single parameter, relaxation-term
leading to constant Prandtl numbers. The polynomial expansion used for the
equilibrium distribution introduces an upper-bound in the local macroscopic
speed. Most widely used time discretization procedures give an explicit
numerical scheme with second-order time step errors. In thermal problems,
quadrature did not succeed in giving discrete velocity sets able to generate
multi-speed regular lattices. All these problems, greatly, difficult the
numerical simulation of LBM based algorithms. In present work, the systematic
derivation of lattice-Boltzmann models from the continuous Boltzmann equation
is discussed. The collision term in the linearized Boltzmann equation is
modeled by expanding the distribution function in Hermite tensors.
Thermohydrodynamic macroscopic equations are correctly retrieved with a
second-order model. Velocity discretization is the most critical step in
establishing regular-lattices framework. In the quadrature process, it is shown
that the integrating variable has an important role in defining the equilibrium
distribution and the lattice-Boltzmann model, leading, alternatively, to
temperature dependent velocities (TDV) and to temperature dependent weights
(TDW) lattice-Boltzmann models.
