Dissipativeness of the hyperbolic quadrature method of moments for kinetic equations
Ruixi Zhang, Yihong Chen, Qian Huang, Wen-An Yong
TL;DR
This work provides a rigorous analytic foundation for HyQMOM closures applied to the 1D BGK equation by proving strict hyperbolicity for all orders $n$ through a polynomial-based closure and a factorization of the characteristic polynomial. It further establishes the dissipativeness of the HyQMOM-induced moment system by verifying Yong's structural stability condition, leveraging affine invariance and the theory of orthogonal polynomials associated with realizable moments. The results yield realizability-preserving numerical schemes under CFL-type conditions and clarify the parameter regime (notably $ extgamma=1$) under which affine invariance holds. Together, these findings extend the mathematical understanding of HyQMOM and provide a framework potentially applicable to broader EQMOM variants and higher-dimensional settings.
Abstract
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.