Many macroscopic equations are proposed to describe the rarefied gas dynamics beyond the Navier-Stokes level, either from the mesoscopic Boltzmann equation or some physical arguments, including (i) Burnett, Woods, super-Burnett, augmented Burnett equations derived from the Chapman-Enskog expansion of the Boltzmann equation, (ii) Grad 13, regularized 13/26 moment equations, rational extended thermodynamics equations, and generalized hydrodynamic equations, where the velocity distribution function is expressed in terms of low-order moments and Hermite polynomials, and (iii) bi-velocity equations and “thermo-mechanically consistent" Burnett equations based on the argument of “volume diffusion”. This paper is dedicated to assess the accuracy of these macroscopic equations. We first consider the Rayleigh-Brillouin scattering, where light is scattered by the density fluctuation in gas. In this specific problem macroscopic equations can be linearized and solutions can always be obtained, no matter whether they are stable or not. Moreover, the accuracy assessment is not contaminated by the gas-wall boundary condition in this periodic problem. Rayleigh-Brillouin spectra of the scattered light are calculated by solving the linearized macroscopic equations and compared to those from the linearized Boltzmann equation. We find that (i) the accuracy of Chapman-Enskog expansion does not always increase with the order of expansion, (ii) for the moment method, the more moments are included, the more accurate the results are, and (iii) macroscopic equations based on “volume diffusion" do not work well even when the Knudsen number is very small. Therefore, among about a dozen tested equations, the regularized 26 moment equations are the most accurate. However, for moderate and highly rarefied gas flows, huge number of moments should be included, as the convergence to true solutions is rather slow. The same conclusion is drawn from the problem of sound propagation between the transducer and receiver. This slow convergence of moment equations is due to the incapability of Hermite polynomials in the capturing of large discontinuities and rapid variations of the velocity distribution function. This study sheds some light on how to choose/develop macroscopic equations for rarefied gas dynamics.

中文翻译：

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线性化稀薄气体流宏观方程的精度

气体的密度波动会散射光。在这个特定的问题中，无论宏观方程是稳定的还是不稳定的，宏观方程都可以线性化，并且总是可以得到解。此外，在该周期性问题中，准确性评估不受气体壁边界条件的污染。通过求解线性化的宏观方程，计算散射光的瑞利-布里渊光谱，并与线性化的玻耳兹曼方程进行比较。我们发现（i）Chapman-Enskog展开的精度并不总是随展开的顺序而增加；（ii）矩量法包括的力矩越多，结果就越准确；以及（iii）宏观方程即使Knudsen数很小，基于“体积扩散”的方法也不能很好地工作。在大约十二个测试方程中，正则化的26个矩方程最精确。但是，对于中等和高度稀疏的气体流，应包括大量的矩，因为收敛到真实解的速度很慢。从换能器和接收器之间的声音传播问题可以得出相同的结论。矩方程的这种缓慢收敛是由于Hermite多项式无法捕获大的不连续性和速度分布函数的快速变化所致。这项研究为如何选择/开发稀有气体动力学的宏观方程式提供了一些启示。因为真正解决方案的融合相当缓慢。从换能器和接收器之间的声音传播问题可以得出相同的结论。矩方程的这种缓慢收敛是由于Hermite多项式无法捕获大的不连续性和速度分布函数的快速变化所致。这项研究为如何选择/开发稀有气体动力学的宏观方程式提供了一些启示。因为真正解决方案的融合相当缓慢。从换能器和接收器之间的声音传播问题可以得出相同的结论。矩方程的这种缓慢收敛是由于Hermite多项式无法捕获大的不连续性和速度分布函数的快速变化所致。这项研究为如何选择/开发稀有气体动力学的宏观方程式提供了一些启示。