Deriving thermal lattice-Boltzmann models from the continuous Boltzmann   equation: theoretical aspects
  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. 
Paulo Cesar Philippi, Luiz Adolfo Hegele Jr., Luis Orlando Emerich dos   Santos and Rodrigo Surmas