Positive filtered Pn moment closures for linear kinetic equations

M. Paul Laiu, Cory D. Hauck, Ryan G. McClarren, Dianne P. O'Leary, André L. Tits

Research output: Contribution to journalArticlepeer-review

16 Scopus citations

Abstract

We propose a positive-preserving moment closure for linear kinetic transport equations based on a filtered spherical harmonic (FPN) expansion in the angular variable. The recently proposed FPN moment equations are known to suffer from the occurrence of (unphysical) negative particle concentrations. The origin of this problem is that the FPN approximation is not always positive at the kinetic level; the new FPN + closure is developed to address this issue. A new spherical harmonic expansion is computed via the solution of an optimization problem, with constraints that enforce positivity, but only on a finite set of preselected points. Combined with an appropriate PDE solver for the moment equations, this ensures positivity of the particle concentration at each step in the time integration. Under an additional, mild regularity assumption, we prove that FPN + has the same consistency as FPN; that is, the FP+ N approximation converges to a given target function in L2 at the same rate as the FPN approximation. Numerical tests llkjklklsuggest that this additional assumption may not be necessary. We also simulate the challenging line source benchmark problem using several different choices of closure. Among the choices that preserve positivity of the particle concentration, the proposed FPN + closure gives the most accurate solution to the line source problem and does so in the least computational time. In addition, we observe that for a regularized version of the line source problem, the FPN + closure does not affect the space-time convergence rate of the PDE solver.

Original languageEnglish
Pages (from-to)3214-3238
Number of pages25
JournalSIAM Journal on Numerical Analysis
Volume54
Issue number6
DOIs
StatePublished - 2016

Funding

The first author's research was supported by the U.S. Department of Energy under grant DESC0001862 and the SCGSR program administered by the Oak Ridge Institute for Science and Education under contract DE-AC05-06OR23100. The second author's research was sponsored by the Office of Advanced Scientific Computing Research and performed at the Oak Ridge National Laboratory, which is managed by UT-Battelle, LLC, under contract DE-AC05-00OR22725. The third author's research was supported by the National Science Foundation under grant 1217170. The fourth and fifth authors's research was supported by the U.S. Department of Energy under grant DESC0001862.

Keywords

  • Filtering
  • Kinetic equation
  • Moment closure
  • Positivity preserving
  • Spectral methods
  • Spherical harmonic expansion

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