Relating Eddington factors to flux limiters

doi:10.1016/0022-4073(84)90112-2

Journal of Quantitative Spectroscopy and Radiative Transfer

Volume 31, Issue 2, February 1984, Pages 149-160

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Abstract

Variable Eddington factors and flux-limiters have been introduced in the P-1 and diffusion equations, respectively, to handle situations when the underlying intensity is too anisotropic for the unmodified theories to remain valid. We present a derivation of a relation between the two for which a new approach to the diffusion approximation is used. Algebraic expressions for Eddington factors satisfying the moment conditions are not satisfactory for closing the P-1 equations but, by using the derived relation, yield acceptable flux-limited diffusion theories.

References (13)

G.C. Pomraning
JQSRT
(1969)
G.N. Minerbo
JQSRT
(1978)
G.C. Pomraning
Nukleonik
(1965)
D.S. Kershaw
Flux Limiting Nature's Own Way
J.M. LeBlanc et al.
Appl. J.
(1970)
A.M. Winslow
Nucl. Sci. Engng
(1968)

There are more references available in the full text version of this article.

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^☆: This work was performed under the auspices of the U.S. Department of Energy at the Lawrence Livermore National Laboratory under contract number W-7405-ENG-48.

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Relating Eddington factors to flux limiters☆

Abstract

JQSRT

JQSRT

Nukleonik

Flux Limiting Nature's Own Way

Appl. J.

Nucl. Sci. Engng