# Stability Analysis of Quadrature-based Moment Methods for Kinetic Equations

## TL;DR

This paper provides a theoretical stability analysis of quadrature-based moment methods for kinetic equations, showing conditions for hyperbolicity and preserving physical properties, which enhances understanding of their applicability.

## Contribution

It offers the first thorough stability analysis of QBMM for Boltzmann equations, identifying conditions for hyperbolicity and the preservation of the H-theorem.

## Key findings

- Gaussian approximation yields hyperbolic moment systems
- Delta-function approximation can lead to non-hyperbolic systems
- The equilibrium manifold is characterized on the boundary of the state space

## Abstract

In this paper, we present a systematic stability analysis of the quadrature-based moment method (QBMM) for the one-dimensional Boltzmann equation with BGK or Shakhov models. As reported in recent literature, the method has revealed its potential for modeling non-equilibrium flows, while a thorough theoretical analysis is largely missing but desirable. We show that the method can yield non-hyperbolic moment systems if the distribution function is approximated by a linear combination of $\delta$-functions. On the other hand, if the $\delta$-functions are replaced by their Gaussian approximations with a common variance, we prove that the moment systems are strictly hyperbolic and preserve the dissipation property (or $H$-theorem) of the kinetic equation. In the proof we also determine the equilibrium manifold that lies on the boundary of the state space. The proofs are quite technical and involve detailed analyses of the characteristic polynomials of the coefficient matrices.