This paper presents a dissipativeness analysis of a quadrature method of moments (called HyQMOM) for the one-dimensional BGK equation. The method has exhibited its good performance in numerous applications. However, its mathematical foundation has not been clarified. Here we present an analytical proof of the strict hyperbolicity of the HyQMOM-induced moment closure systems by introducing a polynomial-based closure technique. As a byproduct, a class of numerical schemes for the HyQMOM system is shown to be realizability preserving under CFL-type conditions. We also show that the system preserves the dissipative properties of the kinetic equation by verifying a certain structural stability condition. The proof uses a newly introduced affine invariance and the homogeneity of the HyQMOM and heavily relies on the theory of orthogonal polynomials associated with realizable moments, in particular, the moments of the standard normal distribution.

翻译：本文对一维BGK方程的一种矩量求积法（称为HyQMOM）进行了耗散性分析。该方法已在众多应用中展现出优异性能，但其数学基础尚未得到阐明。本文通过引入基于多项式的封闭技术，给出了HyQMOM诱导矩封闭系统严格双曲性的解析证明。作为推论，我们证明在CFL型条件下，一类针对HyQMOM系统的数值格式能够保持可实现性。通过验证特定的结构稳定性条件，我们还证明了该系统保持了原动力学方程的耗散特性。证明过程运用了新引入的仿射不变性及HyQMOM的齐次性，并深度依赖于可实现矩量（特别是标准正态分布的矩量）相关的正交多项式理论。