A Framework of Model Reduction with Arbitrary Orders of Accuracy for the Boltzmann Equation
Zhenning Cai, Ruo Li, Yixiao Lu, Yanli Wang
TL;DR
This work develops a general framework for reduced Boltzmann models with arbitrary Knudsen-number accuracy across general collision models by constructing nested, orthogonally defined function spaces and projecting the Boltzmann equation onto them. It proves that the projection retains all information through order $O(\mathrm{Kn}^n)$ and yields hyperbolic and regularized reduced models that achieve $O(\mathrm{Kn}^{n+1})$ accuracy (and $O(\mathrm{Kn}^{2n})$ for linearized collisions in the hyperbolic case). The framework is then instantiated to derive Burnett- and super-Burnett-order 13-moment systems, including regularized variants, for arbitrary collision models; for Maxwell molecules these reduce to Grad’s classic 13-moment equations or Struchtrup’s regularized form, respectively. This approach offers stable, higher-order kinetic models with manageable dimensionality, enabling accurate simulations across transitional to near-continuum regimes. The results provide a principled path from kinetic theory to tractable, accurate macroscopic closures for a broad class of collisional dynamics.
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.