Nonnegative methods for bilinear discontinuous differencing of the S _{N} equations on quadrilaterals
Historically, matrix lumping and ad hoc flux fixups have been the only methods used to eliminate or suppress negative angular flux solutions associated with the unlumped bilinear discontinuous (UBLD) finite element spatial discretization of the twodimensional S _{N} equations. Though matrix lumping inhibits negative angular flux solutions of the S _{N} equations, it does not guarantee strictly positive solutions. In this paper, we develop and define a strictly nonnegative, nonlinear, PetrovGalerkin finite element method that fully preserves the bilinear discontinuous spatial moments of the transport equation. Additionally, we define two ad hoc fixups that maintain particle balance and explicitly set negative nodes of the UBLD finite element solution to zero but use different auxiliary equations to fully define their respective solutions. We assess the ability to inhibit negative angular flux solutions and the accuracy of every spatial discretization that we consider using a glancing void test problem with a discontinuous solution known to stress numerical methods. Though significantly more computationally intense, the nonlinear PetrovGalerkin scheme results in a strictly nonnegative solution and is a more accurate solution than all the other methods considered. One fixup, based on shape preserving, results in a strictly nonnegative final solution but has increased numericalmore »
 Authors:

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 Lawrence Livermore National Lab. (LLNL), Livermore, CA (United States)
 Texas A & M Univ., College Station, TX (United States)
 Publication Date:
 Report Number(s):
 LLNLJRNL697017
Journal ID: ISSN 00295639; TRN: US1701103
 Grant/Contract Number:
 AC5207NA27344
 Type:
 Accepted Manuscript
 Journal Name:
 Nuclear Science and Engineering
 Additional Journal Information:
 Journal Volume: 185; Journal Issue: 1; Journal ID: ISSN 00295639
 Publisher:
 American Nuclear Society
 Research Org:
 Lawrence Livermore National Lab. (LLNL), Livermore, CA (United States)
 Sponsoring Org:
 USDOE
 Country of Publication:
 United States
 Language:
 English
 Subject:
 22 GENERAL STUDIES OF NUCLEAR REACTORS; 97 MATHEMATICS, COMPUTING, AND INFORMATION SCIENCE; radiation transport; DFEM; nonnegative; nonlinear; fixup
 OSTI Identifier:
 1343843
Maginot, Peter G., Ragusa, Jean C., and Morel, Jim E.. Nonnegative methods for bilinear discontinuous differencing of the SN equations on quadrilaterals. United States: N. p.,
Web. doi:10.13182/NSE1638.
Maginot, Peter G., Ragusa, Jean C., & Morel, Jim E.. Nonnegative methods for bilinear discontinuous differencing of the SN equations on quadrilaterals. United States. doi:10.13182/NSE1638.
Maginot, Peter G., Ragusa, Jean C., and Morel, Jim E.. 2016.
"Nonnegative methods for bilinear discontinuous differencing of the SN equations on quadrilaterals". United States.
doi:10.13182/NSE1638. https://www.osti.gov/servlets/purl/1343843.
@article{osti_1343843,
title = {Nonnegative methods for bilinear discontinuous differencing of the SN equations on quadrilaterals},
author = {Maginot, Peter G. and Ragusa, Jean C. and Morel, Jim E.},
abstractNote = {Historically, matrix lumping and ad hoc flux fixups have been the only methods used to eliminate or suppress negative angular flux solutions associated with the unlumped bilinear discontinuous (UBLD) finite element spatial discretization of the twodimensional SN equations. Though matrix lumping inhibits negative angular flux solutions of the SN equations, it does not guarantee strictly positive solutions. In this paper, we develop and define a strictly nonnegative, nonlinear, PetrovGalerkin finite element method that fully preserves the bilinear discontinuous spatial moments of the transport equation. Additionally, we define two ad hoc fixups that maintain particle balance and explicitly set negative nodes of the UBLD finite element solution to zero but use different auxiliary equations to fully define their respective solutions. We assess the ability to inhibit negative angular flux solutions and the accuracy of every spatial discretization that we consider using a glancing void test problem with a discontinuous solution known to stress numerical methods. Though significantly more computationally intense, the nonlinear PetrovGalerkin scheme results in a strictly nonnegative solution and is a more accurate solution than all the other methods considered. One fixup, based on shape preserving, results in a strictly nonnegative final solution but has increased numerical diffusion relative to the PetrovGalerkin scheme and is less accurate than the UBLD solution. The second fixup, which preserves as many spatial moments as possible while setting negative values of the unlumped solution to zero, is less accurate than the PetrovGalerkin scheme but is more accurate than the other fixup. However, it fails to guarantee a strictly nonnegative final solution. As a result, the fully lumped bilinear discontinuous finite element solution is the least accurate method, with significantly more numerical diffusion than the PetrovGalerkin scheme and both fixups.},
doi = {10.13182/NSE1638},
journal = {Nuclear Science and Engineering},
number = 1,
volume = 185,
place = {United States},
year = {2016},
month = {12}
}