Role of higher-order Hermite polynomials in the central-moments-based lattice Boltzmann framework

The cascaded lattice Boltzmann method decomposes the collision stage on a basis of central moments on which the equilibrium state is assumed equal to that of the continuous Maxwellian distribution. Such a relaxation process is usually considered as an assumption, which is then justified a posteriori by showing the enhanced Galilean invariance of the resultant algorithm. An alternative method is to relax central moments to the equilibrium state of the discrete second-order truncated distribution. In this paper, we demonstrate that relaxation to the continuous Maxwellian distribution is equivalent to the discrete counterpart if higher-order (up to sixth) Hermite polynomials are used to construct the equilibrium when the D3Q27 lattice velocity space is considered. Therefore, a theoretical a priori justification of the choice of the continuous distribution is formally provided for the first time.