# Generalized lattice Boltzmann method: Modeling, analysis, and elements

## TL;DR

This paper introduces a unified framework for the generalized lattice Boltzmann method (GLBM), compares four analysis techniques, and discusses implementation elements, showing their equivalence and how existing models fit into this framework.

## Contribution

It provides a comprehensive, unified approach to modeling and analyzing GLBM, clarifies the relationship between different analysis methods, and offers practical elements for implementation.

## Key findings

- Four analysis methods are mathematically equivalent.
- Existing LB models can be derived from the proposed GLBM.
- The framework simplifies the understanding and development of LB models.

## Abstract

In this paper, we first present a unified framework for the modelling of generalized lattice Boltzmann method (GLBM). We then conduct a comparison of the four popular analysis methods (Chapman-Enskog analysis, Maxwell iteration, direct Taylor expansion and recurrence equations approaches) that have been used to obtain the macroscopic Navier-Stokes equations and nonlinear convection-diffusion equations from the GLBM, and show that from mathematical point of view, these four analysis methods are equivalent to each other. Finally, we give some elements that are needed in the implementation of the GLBM, and also find that some available LB models can be obtained from this GLBM.