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Title: A flexible nonlinear diffusion acceleration method for the S N transport equations discretized with discontinuous finite elements

This paper presents a flexible nonlinear diffusion acceleration (NDA) method that discretizes both the S N transport equation and the diffusion equation using the discontinuous finite element method (DFEM). The method is flexible in that the diffusion equation can be discretized on a coarser mesh with the only restriction that it is nested within the transport mesh and the FEM shape function orders of the two equations can be different. The consistency of the transport and diffusion solutions at convergence is defined by using a projection operator mapping the transport into the diffusion FEM space. The diffusion weak form is based on the modified incomplete interior penalty (MIP) diffusion DFEM discretization that is extended by volumetric drift, interior face, and boundary closure terms. In contrast to commonly used coarse mesh finite difference (CMFD) methods, the presented NDA method uses a full FEM discretized diffusion equation for acceleration. Suitable projection and prolongation operators arise naturally from the FEM framework. Via Fourier analysis and numerical experiments for a one-group, fixed source problem the following properties of the NDA method are established for structured quadrilateral meshes: (1) the presented method is unconditionally stable and effective in the presence of mild material heterogeneities ifmore » the same mesh and identical shape functions either of the bilinear or biquadratic type are used, (2) the NDA method remains unconditionally stable in the presence of strong heterogeneities, (3) the NDA method with bilinear elements extends the range of effectiveness and stability by a factor of two when compared to CMFD if a coarser diffusion mesh is selected. In addition, the method is tested for solving the C5G7 multigroup, eigenvalue problem using coarse and fine mesh acceleration. Finally, while NDA does not offer an advantage over CMFD for fine mesh acceleration, it reduces the iteration count required for convergence by almost a factor of two in the case of coarse mesh acceleration.« less
Authors:
 [1] ;  [1] ;  [1] ;  [1] ;  [1] ;  [1] ;  [1] ;  [1] ;  [1]
  1. Idaho National Lab. (INL), Idaho Falls, ID (United States)
Publication Date:
Report Number(s):
INL/JOU-16-37745
Journal ID: ISSN 0021-9991; PII: S0021999117301286
Grant/Contract Number:
AC07-05ID14517
Type:
Accepted Manuscript
Journal Name:
Journal of Computational Physics
Additional Journal Information:
Journal Volume: 338; Journal ID: ISSN 0021-9991
Publisher:
Elsevier
Research Org:
Idaho National Lab. (INL), Idaho Falls, ID (United States)
Sponsoring Org:
USDOE Office of Nuclear Energy (NE)
Country of Publication:
United States
Language:
English
Subject:
73 NUCLEAR PHYSICS AND RADIATION PHYSICS; neutron transport equation; nonlinear diffusion acceleration; discontinuous finite element method
OSTI Identifier:
1363728
Alternate Identifier(s):
OSTI ID: 1397847

Schunert, Sebastian, Wang, Yaqi, Gleicher, Frederick, Ortensi, Javier, Baker, Benjamin, Laboure, Vincent, Wang, Congjian, DeHart, Mark, and Martineau, Richard. A flexible nonlinear diffusion acceleration method for the SN transport equations discretized with discontinuous finite elements. United States: N. p., Web. doi:10.1016/j.jcp.2017.01.070.
Schunert, Sebastian, Wang, Yaqi, Gleicher, Frederick, Ortensi, Javier, Baker, Benjamin, Laboure, Vincent, Wang, Congjian, DeHart, Mark, & Martineau, Richard. A flexible nonlinear diffusion acceleration method for the SN transport equations discretized with discontinuous finite elements. United States. doi:10.1016/j.jcp.2017.01.070.
Schunert, Sebastian, Wang, Yaqi, Gleicher, Frederick, Ortensi, Javier, Baker, Benjamin, Laboure, Vincent, Wang, Congjian, DeHart, Mark, and Martineau, Richard. 2017. "A flexible nonlinear diffusion acceleration method for the SN transport equations discretized with discontinuous finite elements". United States. doi:10.1016/j.jcp.2017.01.070. https://www.osti.gov/servlets/purl/1363728.
@article{osti_1363728,
title = {A flexible nonlinear diffusion acceleration method for the SN transport equations discretized with discontinuous finite elements},
author = {Schunert, Sebastian and Wang, Yaqi and Gleicher, Frederick and Ortensi, Javier and Baker, Benjamin and Laboure, Vincent and Wang, Congjian and DeHart, Mark and Martineau, Richard},
abstractNote = {This paper presents a flexible nonlinear diffusion acceleration (NDA) method that discretizes both the SN transport equation and the diffusion equation using the discontinuous finite element method (DFEM). The method is flexible in that the diffusion equation can be discretized on a coarser mesh with the only restriction that it is nested within the transport mesh and the FEM shape function orders of the two equations can be different. The consistency of the transport and diffusion solutions at convergence is defined by using a projection operator mapping the transport into the diffusion FEM space. The diffusion weak form is based on the modified incomplete interior penalty (MIP) diffusion DFEM discretization that is extended by volumetric drift, interior face, and boundary closure terms. In contrast to commonly used coarse mesh finite difference (CMFD) methods, the presented NDA method uses a full FEM discretized diffusion equation for acceleration. Suitable projection and prolongation operators arise naturally from the FEM framework. Via Fourier analysis and numerical experiments for a one-group, fixed source problem the following properties of the NDA method are established for structured quadrilateral meshes: (1) the presented method is unconditionally stable and effective in the presence of mild material heterogeneities if the same mesh and identical shape functions either of the bilinear or biquadratic type are used, (2) the NDA method remains unconditionally stable in the presence of strong heterogeneities, (3) the NDA method with bilinear elements extends the range of effectiveness and stability by a factor of two when compared to CMFD if a coarser diffusion mesh is selected. In addition, the method is tested for solving the C5G7 multigroup, eigenvalue problem using coarse and fine mesh acceleration. Finally, while NDA does not offer an advantage over CMFD for fine mesh acceleration, it reduces the iteration count required for convergence by almost a factor of two in the case of coarse mesh acceleration.},
doi = {10.1016/j.jcp.2017.01.070},
journal = {Journal of Computational Physics},
number = ,
volume = 338,
place = {United States},
year = {2017},
month = {2}
}