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Scale-induced closure for approximations of kinetic equations

by P. Kauf and M. Torrilhon and M. Junk

(Report number 2010-32)

Abstract
The order-of-magnitude method proposed by Struchtrup (Phys. Fluids 16/11, 2004, p.3921-3934) is a new closure procedure for the infinite moment hierarchy in kinetic theory of gases, taking into account the scaling of the moments. The scaling parameter is the Knudsen number Kn, which is the mean free path of a particle divided by the system size. In this paper, we generalize the order-of-magnitude method and derive a formal theory of scale-induced closures on the level of the kinetic equation. Generally, different orders of magnitude appear through balancing the stiff production term of order 1/Kn with the advection part of the kinetic equation. A cascade of scales is then induced by different powers of Kn. The new closure produces a moment distribution function that respects the scaling of a Chapman-Enskog expansion. The collision operator induces a decomposition of the non-equilibrium part of the distribution function in terms of the Knudsen number. The first iteration of the new closure can be shown to be of second-order in Kn under moderate conditions on the collision operator, to be L^{2} -stable and to possess an entropy law. The derivation of higher order approximations is also possible. We illustrate the features of this approach in the framework of a 16 discrete velocities model.

Keywords:

@Techreport{KTJ10_435,
  author = {P. Kauf and M. Torrilhon and M. Junk},
  title = {Scale-induced closure for approximations of kinetic equations},
  institution = {Seminar for Applied Mathematics, ETH Z{\"u}rich},
  number = {2010-32},
  address = {Switzerland},
  url = {https://www.sam.math.ethz.ch/sam_reports/reports_final/reports2010/2010-32.pdf },
  year = {2010}
}

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