Factorization symmetry in lattice Boltzmann simulations

TL;DR

This paper develops a non-perturbative algebraic theory for lattice Boltzmann methods based on symmetry, enabling systematic derivation and analysis of lattice models in multiple dimensions.

Contribution

It introduces a symmetry-based algebraic framework that allows automated derivation of lattice Boltzmann models and unifies various known lattice configurations.

Findings

Derivation of admissible lattices in 1D using a matching condition.

Analytical form of quasi-equilibrium distribution functions in 2D and 3D.

Automated method for deriving lattice Boltzmann models from 2D tables.

Abstract

A non-perturbative algebraic theory of lattice Boltzmann method is developed based on a symmetry of a product. It involves three steps: (i) Derivation of admissible lattices in one spatial dimension through a matching condition which imposes restricted extension of higher-order Gaussian moments, (ii) Special quasi-equilibrium distribution function found analytically in closed form on the product-lattice in two and three spatial dimensions, and which proves factorization of quasi-equilibrium moments, and (iii) Algebraic method of pruning based on a one-into-one relation between groups of discrete velocities and moments. Two routes of constructing lattice Boltzmann equilibria are distinguished. Present theory includes previously known limiting and special cases of lattices, and enables automated derivation of lattice Boltzmann models from two-dimensional tables, by finding roots of one polynomial and solving a few linear systems.