On the derivations of lattice Boltzmann evolution equation

TL;DR

This paper provides a detailed comparison of three characteristic-line schemes for deriving the evolution equation in lattice Boltzmann methods: Boesch-Karlin (BK), Taylor-expansion (TE), and He-Luo (HL). By tracing their mathematical mechanisms, including Euler–Maclaurin integration, partial integration with Bernoulli polynomials, and integrating-factor methods, the authors reveal that BK, TE, and HL are equivalent under a continuous $g(t')$ along characteristics, yet differ in assumptions about $g(t')$ and their numerical behavior. The analysis highlights preconditions that must be satisfied for meaningful evolution equations, discusses reach and limitations of each approach, and provides guidance on the most appropriate scheme for a given collision model and application scenario. The work underscores that while HL is often best for BGK-based evolution, BK and TE offer flexibility and potentially faster convergence when generalized collision models are used, illustrating the trade-offs between analytical exactness and numerical stability. Overall, the paper clarifies the conditions under which these schemes align and how their derivations influence their applicability and future development in LBM theory.

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.