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

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2021-01-01
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Fox, Rodney
Laurent, Frédérique
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Chemical and Biological Engineering
Abstract
A tractable solution is proposed to a classical problem in kinetic theory, namely, givenany set of realizable velocity moments up to order 2n, a closure for the moment of order 2n+ 1 isconstructed 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 hyperbolicquadrature method of moments (HyQMOM) was introduced to close this moment system up to fourthorder (n≤2). Here, HyQMOM is reformulated and extended to arbitrary even-order moments. TheHyQMOM closure is defined based on the properties of the monic orthogonal polynomialsQnthatare uniquely defined by the velocity moments up to order 2n−1. Thus, HyQMOM is strictly amoment closure and does not rely on the reconstruction of a velocity distribution function with thesame moments. On the boundary of moment space,ndouble roots of the characteristic polynomialP2n+1of the Jacobian matrix of the system are the roots ofQn, while in the interior,P2n+1andQnsharenroots. The remainingn+ 1 roots ofP2n+1bound and separate the roots ofQn. An efficientalgorithm, based on the Chebyshev algorithm, for computing the moment of order 2n+ 1 from themoments up to order 2nis developed. The analytical solution to a 1-D Riemann problem is used todemonstrate convergence of the HyQMOM closure with increasingn. .
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This is a manuscript of an article published as Fox, Rodney O., and Frédérique Laurent. "Hyperbolic quadrature method of moments for the one-dimensional kinetic equation." SIAM Journal on Applied Mathematics 82, no. 2 (2022): 750-771. DOI: 10.1137/21M1406143. Attribution 4.0 International (CC BY 4.0). Copyright 2022 Society for Industrial and Applied Mathematics. Posted with permission.
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Fri Jan 01 00:00:00 UTC 2021
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