# Comprehensive comparison of collision models in the lattice Boltzmann framework: Theoretical investigations

## TL;DR

This paper provides a unified mathematical framework for various collision models in the lattice Boltzmann method, comparing their mathematical properties and impacts on simulation accuracy across multiple lattice configurations.

## Contribution

It introduces a common formalism for diverse collision models, analyzes their relationships, and evaluates the effects of different polynomial bases and regularization on LBMs.

## Key findings

- Unified formalism for collision models in LBM
- Impact of polynomial basis and regularization on accuracy
- Links established between different collision models

## Abstract

Over the last decades, several types of collision models have been proposed to extend the validity domain of the lattice Boltzmann method (LBM), each of them being introduced in its own formalism. The present article proposes a formalism that describes all these methods within a common mathematical framework, and in this way allows us to draw direct links between them. Here, the focus is put on single and multirelaxation time collision models in either their raw moment, central moment, cumulant or regularized form. In parallel with that, several bases (non orthogonal, orthogonal, Hermite) are considered for the polynomial expansion of populations. General relationships between moments are first derived to understand how moment spaces are related to each other. In addition, a review of collision models further sheds light on collision models that can be rewritten in a linear matrix form. More quantitative mathematical studies are then carried out by comparing explicit expressions for the post collision populations. Thanks to this, it is possible to deduce the impact of both the polynomial basis (raw, Hermite, central, central Hermite, cumulant) and the inclusion of regularization steps on isothermal LBMs. Extensive results are provided for the D1Q3, D2Q9, and D3Q27 lattices, the latter being further extended to the D3Q19 velocity discretization. Links with the most common two and multirelaxation time collision models are also provided for the sake of completeness. The present work ends by emphasizing the importance of an accurate representation of the equilibrium state, independently of the choice of moment space. As an addition to the theoretical purpose of the present article, general instructions are provided to help the reader with the implementation of the most complicated collision models.