# On-node lattices construction using $\textit{partial}$ Gauss-Hermite quadrature for the lattice Boltzmann method

## TL;DR

This paper introduces the partial Gauss-Hermite quadrature (pGHQ), a concise and unified method for constructing on-node lattices in the lattice Boltzmann method, enabling optimal lattice searches and stability improvements.

## Contribution

The paper develops the pGHQ framework, simplifying lattice construction, unifying symmetric and asymmetric lattices, and enabling full-range quadrature degree coverage for LB models.

## Key findings

- Identified a wide variety of available lattices for different moment degrees.
- Discrete velocity set significantly affects the positivity of equilibrium distributions.
- pGHQ provides a foundation for enhancing LB stability through velocity set modifications.

## Abstract

A concise theoretical framework, the $\textit{partial}$ Gauss-Hermite quadrature (pGHQ), is established for constructing on-node lattices of the lattice Boltzmann (LB) method under a Cartesian coordinate system. Comparing with existing approaches, the pGHQ scheme has the following advantages: $\textbf{a).}$ extremely concise algorithm, $\textbf{b).}$ unifying the constructing procedure of symmetric and asymmetric on-node lattices, $\textbf{c).}$ covering full-range quadrature degree of a given discrete velocity set. We employ it to search the local optimal and asymmetric lattices for $\left\{ {n = 3,4,5,6,7} \right\}$ moment degree equilibrium distribution discretization on range $\left[ { - 10,10} \right]$. 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 major impact to the numerical stability of the LB method. Hence the results of the pGHQ scheme lay a foundation for further investigations on improving the numerical stability of the LB method by modifying the discrete velocity set. It also worths noting that pGHQ can be extended into the entropic LB model though it was proposed for the Hermite polynomial expansion LB theory.