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# Optimal prediction for moment models: Crescendo diffusion and reordered equations

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### Abstract

A direct numerical solution of the radiative transfer equation or any kinetic equation is typically expensive, since the radiative intensity depends on time, space and direction. An expansion in the direction variables yields an equivalent system of infinitely many moments. A fundamental problem is how to truncate the system. Various closures have been presented in the literature. We want to study moment closure generally within the framework of optimal prediction, a strategy to approximate the mean solution of a large system by a smaller system, for radiation moment systems. We apply this strategy to radiative transfer and show that several closures can be re-derived within this framework, e.g. $$P_N$$, diffusion, and diffusion correction closures. In addition, the formalism gives rise to new parabolic systems, the reordered $$P_N$$ equations, that are similar to the simplified $$P_N$$ equations. Furthermore, we propose a modification to existing closures. Although simple and with no extra cost, this newly derived crescendo diffusion yields better approximations in numerical tests.

### Author and article information

###### Journal
2009-02-02
2009-06-13
###### Article
0902.0076
e5e45f44-9b67-4363-82b0-d2006e690ea2

76P05, 62Pxx
Continuum Mech. Thermodyn. 21 (2009) 511
Revised version: 17 pages, 6 figures, presented at Workshop on Moment Methods in Kinetic Gas Theory, ETH Zurich, 2008 2 figures added, minor corrections
math-ph math.MP

Mathematical physics,Mathematical & Computational physics
Mathematical physics, Mathematical & Computational physics