Hyperbolic quadrature method of moments for the one-dimensional kinetic equation

Abstract

A solution is proposed to a longstanding open 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 Qn 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 P2n+1 are the roots of Qn, while in the interior, P 2n+1 and Qn share n roots. The remaining n + 1 roots of P2n+1 bound and separate the roots of Qn. 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.