We propose a diagram notation for the derivation of hyperbolic moment models for the Boltzmann equation that yields a better understanding of the resulting moment systems. So far several hyperbolic moment models were presented, but their derivations are often very technical and there is little insight into the explicit form of the equations. In our diagram notation, each term in the moment equations can be explicitly tracked throughout the derivation process and whether the resulting moment system is hyperbolic can be easily observed from the diagram. We apply the diagram notation to derive existing moment models, including Grad’s moment equations, hyperbolic moment equations, quadrature-based moment equations, and explain differences among them. Moreover, a system called Simplified Hyperbolic Moment Equations (SHME) is also derived by the diagram notation. Numerical tests of a 1D shock tube show that the simplifications are too strong to obtain convergence for a large number of moments but the simplified model gives accurate solutions for small number of moments.

中文翻译：

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双曲矩系统推导的图表符号

我们为玻尔兹曼方程的双曲矩模型的推导提出了一种图表符号，它可以更好地理解所产生的矩系统。到目前为止，提出了几个双曲矩模型，但它们的推导通常非常技术性，并且对方程的显式形式知之甚少。在我们的图表符号中，可以在整个推导过程中明确跟踪力矩方程中的每一项，并且可以从图表中轻松观察到所产生的力矩系统是否是双曲线的。我们应用图表符号来推导现有的矩模型，包括 Grad 矩方程、双曲矩方程、基于正交的矩方程，并解释它们之间的差异。而且，一个称为简化双曲矩方程 (SHME) 的系统也由图表符号导出。一维激波管的数值试验表明，对于大量矩，简化太强而无法收敛，但简化模型对少量矩给出了准确的解。