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

dc.contributorMorel, Jim E.
dc.contributorRagusa, Jean C.
dc.creatorMaginot, Peter Gregory
dc.date.accessioned2012-02-14T22:19:00Z
dc.date.accessioned2012-02-16T16:12:51Z
dc.date.accessioned2017-04-07T19:59:00Z
dc.date.available2012-02-14T22:19:00Z
dc.date.available2012-02-16T16:12:51Z
dc.date.available2017-04-07T19:59:00Z
dc.date.created2010-12
dc.date.issued2012-02-14
dc.description.abstractLinear 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.
dc.identifier.urihttp://hdl.handle.net/1969.1/ETD-TAMU-2010-12-8976
dc.language.isoen_US
dc.subjectStrictly positive closure
dc.subjectDiscrete ordinates method
dc.subjectRadiation transport
dc.subjectDiscontinuous finite elements
dc.titleA Nonlinear Positive Extension of the Linear Discontinuous Spatial Discretization of the Transport Equation
dc.typeThesis

Files