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Chemical and Biological Engineering

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Advanced and Nanostructured Materials, Computational Fluid Dynamics

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A solution is proposed to a longstanding open problem in kinetic theory, namely, 5 given any set of realizable velocity moments up to order 2n, a closure for the moment of order 2n+1 is 6 constructed for which the moment system found from the free-transport term in the one-dimensional 7 (1-D) kinetic equation is globally hyperbolic and in conservative form. In prior work, the hyperbolic 8 quadrature method of moments (HyQMOM) was introduced to close this moment system up to fourth 9 order (n ≤ 2). Here, HyQMOM is reformulated and extended to arbitrary even-order moments. The 10 HyQMOM closure is defined based on the properties of the monic orthogonal polynomials Qn that 11 are uniquely defined by the velocity moments up to order 2n − 1. Thus, HyQMOM is strictly a 12 moment closure and does not rely on the reconstruction of a velocity distribution function with the 13 same moments. On the boundary of moment space, n double roots of the characteristic polynomial 14 P2n+1 are the roots of Qn, while in the interior, P2n+1 and Qn share n roots. The remaining 15 n + 1 roots of P2n+1 bound and separate the roots of Qn. An efficient algorithm, based on the 16 Chebyshev algorithm, for computing the moment of order 2n + 1 from the moments up to order 2n 17 is developed. The analytical solution to a 1-D Riemann problem is used to demonstrate convergence 18 of the HyQMOM closure with increasing n.


This is a pre-print of the article Fox, Rodney, and Frédérique Laurent. "Hyperbolic quadrature method of moments for the one-dimensional kinetic equation." arXiv preprint arXiv:2103.10138 (2021). Posted with permission.

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