# On the derivations of lattice Boltzmann evolution equation

## TL;DR

This paper compares different schemes for deriving the evolution equation in lattice Boltzmann methods, analyzing their mathematical foundations, advantages, and limitations to guide their application and future development.

## Contribution

It introduces a new Taylor-expansion scheme and provides a detailed comparative analysis of classical and novel derivation methods in LBM.

## Key findings

- The Taylor-expansion scheme extends the He-Luo scheme.
- Analysis of the mathematical mechanisms behind each scheme.
- Guidelines for selecting schemes based on their derivation and application scenarios.

## Abstract

A comparative analysis on the popular schemes for evaluating evolution equation in lattice Boltzmann method (LBM) is presented in this paper. It includes two classical characteristic-line schemes, Boesh-Karlin and He-Luo scheme, and a author-proposed scheme, Taylor-expansion scheme, originating from the extension of He-Luo scheme. We detailly discuss the mathematical mechanism and the equilibrium distribution evolution behind them. By analyzing the conflict between prediction and derivation, we address the preconditions for these schemes. At the end, we conclude their pros and cons and suggest scheme's applicable scene based on their derivation procedure and further development capacity.