On-node lattices construction using partial Gauss-Hermite quadrature for the lattice Boltzmann method

A concise theoretical framework, the partial Gauss-Hermite quadrature (pGHQ), is established to construct on-node lattices of the lattice Boltzmann (LB) method under a Cartesian coordinate system. Compared with the existing approaches, the pGHQ scheme has the following advantages: extremely concise algorithm, unifies the constructing procedure for symmetric and asymmetric on-node lattices, and covers a full-range quadrature degree of a given discrete velocity set. We employ the pGHQ scheme to search the local optimal and asymmetric lattices for {n = 3,4,5, 6,7} moment degree equilibrium distribution discretization on the range [-10,10]. The search reveals a surprising abundance of available lattices. Through a brief analysis, the discrete velocity set shows a significant influence on the positivity of equilibrium distributions, which is considered as one of the major impacts of the numerical stability of the LB method. Hence, the results of the pGHQ scheme lay a foundation for further investigations to improve the numerical stability of the LB method by modifying the discrete velocity set. It is also worth noting that pGHQ can be extended into the entropic LB model, even though it was proposed for the Hermite polynomial expansion LB theory.