Dissipativeness of the hyperbolic quadrature method of moments for kinetic equations

TL;DR

This paper provides a mathematical foundation for the HyQMOM method for kinetic equations, proving its hyperbolicity, realizability, and dissipative properties using polynomial techniques and invariance principles.

Contribution

It introduces an analytical proof of HyQMOM's hyperbolicity and dissipativity, establishing its theoretical robustness for kinetic equation modeling.

Findings

Proves strict hyperbolicity of HyQMOM-induced systems.

Shows numerical schemes preserve realizability under CFL conditions.

Verifies that HyQMOM maintains the dissipative nature of the kinetic equation.

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.