A Framework of Model Reduction with Arbitrary Orders of Accuracy for the Boltzmann Equation

TL;DR

This paper develops a systematic framework for creating reduced models of the Boltzmann equation with customizable accuracy levels, improving upon traditional methods and deriving higher-order moment systems applicable to various collision models.

Contribution

The paper introduces a novel orthogonal decomposition-based framework for model reduction of the Boltzmann equation with arbitrary accuracy, surpassing the Chapman-Enskog expansion in tractability.

Findings

Reduced models can achieve higher accuracy orders than the retained terms suggest.

Linearized collision models significantly enhance accuracy to the order of $O(\mathit{Kn}^{2n})$.

Explicit derivation of 13-moment Burnett and super-Burnett systems for general collision models.

Abstract

This paper presents a general framework for constructing reduced models that approximate the Boltzmann equation with arbitrary orders of accuracy in terms of the Knudsen number $\mathit{Kn}$, applicable to general collision models in rarefied gas dynamics. The framework is based on an orthogonal decomposition of the distribution function into components of different orders in $\mathit{Kn}$, from which the reduced models are systematically derived through asymptotic analysis. Compared to the Chapman-Enskog expansion, our approach yields more tractable model structures. Notably, we establish that a reduced model retaining all terms up to $O(\mathit{Kn}^n)$ in the expansion surprisingly yields models with order of accuracy $O(\mathit{Kn}^{n+1})$. Furthermore, when the collision term is linearized, the accuracy improves dramatically to $O(\mathit{Kn}^{2n})$. These results extend to regularized models containing second-order derivatives. As concrete applications, we explicitly derive 13-moment systems of Burnett and super-Burnett orders valid for arbitrary collision models.