Hyperbolic Quadrature Method of Moments for the One-Dimensional Kinetic Equation

Abstract

A tractable solution is proposed to a classical problem in kinetic theory, namely, given any set of realizable velocity moments up to order $2n$, a closure for the moment of order $2n+1$ is constructed for which the moment system found from the free-transport term in the one-dimensional (1-D) kinetic equation is globally hyperbolic and in conservative form. In prior work, the hyperbolic quadrature method of moments (HyQMOM) was introduced to close this moment system up to fourth order ($n \le 2$). Here, HyQMOM is reformulated and extended to arbitrary even-order moments. The HyQMOM closure is defined based on the properties of the monic orthogonal polynomials $Q_n$ that are uniquely defined by the velocity moments up to order $2n-1$. Thus, HyQMOM is strictly a moment closure and does not rely on the reconstruction of a velocity distribution function with the same moments. On the boundary of moment space, $n$ double roots of the characteristic polynomial $P_{2n+1}$ of the Jacobian matrix of the system are the roots of $Q_n$, while in the interior, $P_{2n+1}$ and $Q_n$ share $n$ roots. The remaining $n+1$ roots of $P_{2n+1}$ bound and separate the roots of $Q_n$. An efficient algorithm, based on the Chebyshev algorithm, for computing the moment of order $2n+1$ from the moments up to order $2n$ is developed. The analytical solution to a 1-D Riemann problem is used to demonstrate convergence of the HyQMOM closure with increasing $n$.