Nonnegative methods for bilinear discontinuous differencing of the SN equations on quadrilaterals
Abstract
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 two-dimensional 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, Petrov-Galerkin 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 Petrov-Galerkin 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 relativemore »
- Authors:
-
- Lawrence Livermore National Lab. (LLNL), Livermore, CA (United States)
- Texas A & M Univ., College Station, TX (United States)
- Publication Date:
- Research Org.:
- Lawrence Livermore National Laboratory (LLNL), Livermore, CA (United States)
- Sponsoring Org.:
- USDOE
- OSTI Identifier:
- 1343843
- Report Number(s):
- LLNL-JRNL-697017
Journal ID: ISSN 0029-5639; TRN: US1701103
- Grant/Contract Number:
- AC52-07NA27344
- Resource Type:
- Accepted Manuscript
- Journal Name:
- Nuclear Science and Engineering
- Additional Journal Information:
- Journal Volume: 185; Journal Issue: 1; Journal ID: ISSN 0029-5639
- Publisher:
- American Nuclear Society
- Country of Publication:
- United States
- Language:
- English
- Subject:
- 22 GENERAL STUDIES OF NUCLEAR REACTORS; 97 MATHEMATICS, COMPUTING, AND INFORMATION SCIENCE; radiation transport; DFEM; non-negative; non-linear; fix-up
Citation Formats
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., 2016.
Web. doi:10.13182/NSE16-38.
Maginot, Peter G., Ragusa, Jean C., & Morel, Jim E. Nonnegative methods for bilinear discontinuous differencing of the SN equations on quadrilaterals. United States. https://doi.org/10.13182/NSE16-38
Maginot, Peter G., Ragusa, Jean C., and Morel, Jim E. Thu .
"Nonnegative methods for bilinear discontinuous differencing of the SN equations on quadrilaterals". United States. https://doi.org/10.13182/NSE16-38. 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 two-dimensional 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, Petrov-Galerkin 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 Petrov-Galerkin 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 Petrov-Galerkin 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 Petrov-Galerkin 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 Petrov-Galerkin scheme and both fixups.},
doi = {10.13182/NSE16-38},
journal = {Nuclear Science and Engineering},
number = 1,
volume = 185,
place = {United States},
year = {Thu Dec 22 00:00:00 EST 2016},
month = {Thu Dec 22 00:00:00 EST 2016}
}
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