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A Nonlinear Positive Extension of the Linear Discontinuous Spatial Discretization of the Transport Equation

Linear discontinuous (LD) spatial discretization of the transport operator can
generate negative angular flux solutions. In slab geometry, negativities are limited
to optically thick cells. However, in multi-dimension problems, negativities can even
occur in voids. Past attempts to eliminate the negativities associated with LD have
focused on inherently positive solution shapes and ad-hoc fixups. We present a new,
strictly non-negative finite element method that reduces to the LD method whenever
the LD solution is everywhere positive. The new method assumes an angular flux
distribution, e , that is a linear function in space, but with all negativities set-to-
zero. Our new scheme always conserves the zeroth and linear spatial moments of the
transport equation. For these reasons, we call our method the consistent set-to-zero
(CSZ) scheme.
CSZ can be thought of as a nonlinear modification of the LD scheme. When the
LD solution is everywhere positive within a cell, psi csz = psi LD. If psi LD < 0 somewhere
within a cell, psi csz is a linear function psi csz with all negativities set to zero. Applying
CSZ to the transport moment equations creates a nonlinear system of equations
which is solved to obtain a non-negative solution that preserves the moments of the
transport equation. These properties make CSZ unique; it encompasses the desirable
properties of both strictly positive nonlinear solution representations and ad-hoc
fixups. Our test problems indicate that CSZ avoids the slow spatial convergence
properties of past inherently positive solutions representations, is more accurate than ad-hoc fixups, and does not require significantly more computational work to solve
a problem than using an ad-hoc fixup.
Overall, CSZ is easy to implement and a valuable addition to existing transport
codes, particularly for shielding applications. CSZ is presented here in slab and rect-
angular geometries, but is readily extensible to three-dimensional Cartesian (brick)
geometries. To be applicable to other simulations, particularly radiative transfer,
additional research will need to be conducted, focusing on the diffusion limit in
multi-dimension geometries and solution acceleration techniques.

Identiferoai:union.ndltd.org:tamu.edu/oai:repository.tamu.edu:1969.1/ETD-TAMU-2010-12-8976
Date2010 December 1900
CreatorsMaginot, Peter Gregory
ContributorsMorel, Jim E., Ragusa, Jean C.
Source SetsTexas A and M University
Languageen_US
Detected LanguageEnglish
Typethesis, text
Formatapplication/pdf

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