High-order lattice-Boltzmann

Unlike conventional CFD methods, the lattice Boltzmann method (LBM) describes the dynamic behaviour of physical systems in a mesoscopic scale, based on discrete forms of kinetic equations. In addition to the classical collision-propagation scheme in which the physical and velocity spaces are coupled, finite-differences, finite volumes and finite-element schemes have been used for numerically solving the discrete kinetic equations. A major breakthrough in LB theory was the direct derivation of the LB equation from continuous kinetic equations, establishing a systematic link between the kinetic theory and the lattice Boltzmann method and determining the necessary conditions for the discretization of the velocity space. The lattices obtained by this method proved to be stable in flows over a wide range of parameters, by the use of high-order lattice Boltzmann schemes, leading to velocity sets which, when used in a discrete velocity kinetic scheme, ensures accurate recovery of the high-order hydrodynamic moments. This review presents the theoretical background of these kinetic methods. In particular, we focus on high-order discrete forms of the Boltzmann equation suitable for non-ideal fluids and on the lattice-Boltzmann collision-propagation method.