The velocity discretization is a critical step in deriving the lattice Boltzmann (LBE) from the continuous Boltzmann equation. This problem is considered in the present paper, following an alternative approach and giving the minimal discrete velocity sets in accordance with the order of approximation that is required for the LBE with respect to the continuous Boltzmann equation and with the lattice structure. Considering to be the order of the polynomial approximation to the Maxwell-Boltzmann equilibrium distribution, it is shown that solving the discretization problem is equivalent to finding the inner product in the discrete space induced by the inner product in the continuous space that preserves the norm and the orthogonality of the Hermite polynomial tensors in the Hilbert space generated by the functions that map the velocity space onto the real numbers space. As a consequence, it is shown that, for each order N of approximation, the even-parity velocity tensors are isotropic up to rank 2N in the discrete space. The norm and the orthogonality restrictions lead to space-filling lattices with increased dimensionality when compared with presently known lattices. This problem is discussed in relation with a discretization approach based on a finite set of orthogonal functions in the discrete space. Two-dimensional square lattices intended to be used in thermal problems and their respective equilibrium distributions are presented and discussed.

Although several thermal lattice Boltzmann models have been proposed, this method has not yet been shown to be able to describe nonisothermal fully compressible flows in a satisfactory manner, mostly due to the presence of important deviations from the advection-diffusion macroscopic equations and also due to numerical instabilities. In this context, this paper presents a linear stability analysis for some lattice Boltzmann models that were recently derived as discrete forms of the continuous Boltzmann equation [P. C. Philippi, L. A. Hegele, Jr., L. O. E. dos Santos, and R. Surmas, Phys. Rev. E 63, 056702 (2006)], in order to investigate the sources of instability and to find, for each model, the upper and lower limits for the macroscopic variables, between which it is possible to ensure a stable behavior. The results for two-dimensional (2D) lattices with 9, 17, 25, and 37 velocities indicate that increasing the order of approximation of the lattice Boltzmann equation enhances stability. Results are also presented for an athermal 2D nine-velocity model, the accuracy of which has been improved with respect to the standard D2Q9 model, by adding third-order terms in the lattice Boltzmann equation.

A lattice-Boltzmann equation (LBE) is the discrete counterpart of a continuous kinetic model. It can be derived using a Hermite polynomial expansion for the velocity distribution function. Since LBEs are characterized by discrete, finite representations of the microscopic velocity space, the expansion must be truncated and the appropriate order of truncation depends on the hydrodynamic problem under investigation. Here we consider a particular truncation where the non-equilibrium distribution is expanded on a par with the equilibrium distribution, except that the diffusive parts of high-order non-equilibrium moments are filtered, i.e. only the corresponding advective parts are retained after a given rank. The decomposition of moments into diffusive and advective parts is based directly on analytical relations between Hermite polynomial tensors. The resulting, refined regularization procedure leads to recurrence relations where high-order non-equilibrium moments are expressed in terms of low-order ones. The procedure is appealing in the sense that stability can be enhanced without local variation of transport parameters, like viscosity, or without tuning the simulation parameters based on embedded optimization steps. The improved stability properties are here demonstrated using the perturbed double periodic shear layer flow and the Sod shock tube problem as benchmark cases.

The velocity discretization is a critical step in deriving the lattice Boltzmann (LBE) from the Boltzmann equation. The velocity discretization problem was considered in a recent paper (Philippi et al., From the continuous to the lattice Boltzmann equation: the discretization problem and thermal models, Physical Review E 73: 56702, 2006) following a new approach and giving the minimal discrete velocity sets in accordance with the order of approximation that is required for the LBE with respect to the Boltzmann equation. As a consequence, two-dimensional lattices and their respective equilibrium distributions were derived and discussed, considering the order of approximation that was required for the LBE. In the present work, a Chapman-Enskog (CE) analysis is performed for deriving the macroscopic transport equations for the mass, momentum and energy for these lattices. The problem of describing the transfer of energy in fluids is discussed in relation with the order of approximation of the LBE model. Simulation of temperature, pressure and velocity steps are also presented to validate the CE analysis.