Conditional Hyperbolic Quadrature Method of Moments for Kinetic Equations

Rodney O Fox (1) , Frédérique Laurent (2, 3) , Aymeric Vié (2, 3)

1 ISU - Iowa State University
2 EM2C - Laboratoire d'Énergétique Moléculaire et Macroscopique, Combustion
3 FR3487 - Fédération de Mathématiques de l'Ecole Centrale Paris

The conditional quadrature method of moments (CQMOM) was introduced by Yuan and Fox [J. Comput. Phys. 230 (22), 8216–8246 (2011)] to reconstruct a velocity distribution function (VDF) from a finite set of its integer moments. The reconstructed VDF takes the form of a sum of weighted Dirac delta functions in velocity phase space, and provides a closure for the spatial flux term in the corresponding kinetic equation. The CQMOM closure for the flux leads to a weakly hyperbolic system of moment equations. In subsequent work [Chalons et al., Proceed. CTR Sum. Prog. 2010, 347–358 (2010)], the Dirac delta functions were replaced by Gaussian distributions, which make the moment system hyperbolic but at the added cost of dealing with continuous distributions. Here, a hyperbolic version of CQMOM is proposed that uses weighted Dirac delta functions. While the moment set employed for multi-Gaussian and conditional HyQMOM (CHyQMOM) are equivalent, the latter is able to access all of moment space whereas the former cannot (e.g. arbitrary values of the fourth-order velocity moment in 1-D phase space with two nodes). By making use of the properties of CHyQMOM in 2-D phase space, it is possible to control a symmetrical subset of the optimal moments [Fox, Indust. & Engng. Chem. Res. 48 (21), 9686–9696 (2009)]. Furthermore, the moment sets for 2-D problems are smaller for CHyQMOM than in the original CQMOM thanks to a judicious choice of the velocity abscissas in phase space.

Mots clés: quadrature-based moment methods; kinetic equation; conditional quadrature method of moments; hyperbolic moment closures

Domaines: Analyse numérique [math.NA]; Equations aux dérivées partielles [math.AP]; Mécanique des fluides [physics.class-ph]

Rodney O Fox, Frédérique Laurent, Aymeric Vié. Conditional Hyperbolic Quadrature Method of Moments for Kinetic Equations. Journal of Computational Physics, 2018, 365, pp.269-293. ⟨10.1016/j.jcp.2018.03.025⟩. ⟨hal-01632813v2⟩