Closure equation and higher-order moment relations in the Gauss-Hermite lattice Boltzmann method

Moment methods are often used to solve transport problems involving the Boltzmann-BGK equation. Because the moment equations are underdetermined, these methods require an additional "closure equation" that relates higher to lower-order moments. Here, we examine the closure equation and higher-order moment relations implicit in the lattice Boltzmann method (LBM) that use Gauss-Hermite quadrature for their discrete velocity sets. It is shown that the discrete-velocity-set itself defines the closure equation and higher-order moment relations, the precise forms of which are yet to be reported. The general formula we present facilitates the efficient computational evaluation of higher-order moments and provides insight into the operation of LBM that is anticipated to be useful in theoretical analyses of its performance. Derived formulas for two different velocity sets are validated against numerical implementations of the LBM for steady Couette flow.