  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.
