A note on the lattice Boltzmann method beyond the Chapman Enskog limits

TL;DR

This paper provides a non-perturbative analysis of the BGK kinetic equation, explaining why lattice Boltzmann methods can yield semi-quantitative results in finite-Knudsen regimes relevant for microflow simulations.

Contribution

It offers a theoretical explanation for the effectiveness of lattice Boltzmann methods beyond traditional hydrodynamic limits, especially in moderate-Knudsen regimes.

Findings

Lattice Boltzmann methods remain semi-quantitative up to Knudsen number ~1.

The analysis clarifies the applicability of discrete kinetic models in non-hydrodynamic regimes.

Results support recent microflow simulations aligning with kinetic theory.

Abstract

A non-perturbative analysis of the Bhatnagar-Gross-Krook (BGK) model kinetic equation for finite values of the Knudsen number is presented. This analysis indicates why discrete kinetic versions of the BGK equation, and notably the Lattice Boltzmann method, can provide semi-quantitative results also in the non-hydrodynamic, finite-Knudsen regime, up to $Kn\sim {\cal O}(1)$. This may help the interpretation of recent Lattice Boltzmann simulations of microflows, which show satisfactory agreement with continuum kinetic theory in the moderate-Knudsen regime.