Krylov iterative methods applied to multidimensional S[sub n] calculations in the presence of material discontinuities Page: 2 of 20
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Nuclear Mathematical and Computational Sciences: A Century in Review, A Century Anew
Gatlinburg, Tennessee, April 6-11, 2003, on CD-ROM, American Nuclear Society, LaGrange Park, IL (2003)
KRYLOV ITERATIVE METHODS APPLIED TO MULTIDIMENSIONAL
SN CALCULATIONS IN THE PRESENCE OF MATERIAL
James S. Warsa, Todd A. Wareing, Jim E. Morel
Transport Methods Group
Los Alamos National Laboratory
Los Alamos, NM 87545-0001
warsa@ lanl.gov, wareing @ lanl.gov, email@example.com
We show that a Krylov iterative method, preconditioned with DSA, can be used to efficiently compute
solutions to diffusive problems with discontinuities in material properties. We consider a lumped, linear
discontinuous discretization of the SN transport equation with a "partially consistent" DSA
preconditioner. The Krylov method can be implemented in terms of the original SN source iteration
coding with little modification. Results from numerical experiments show that replacing source
iteration with a preconditioned Krylov method can efficiently solve problems that are virtually
intractable with accelerated source iteration.
Key Words: Krylov iterative methods, discrete ordinates, deterministic transport methods, diffusion
A spatial discretization of the DSA diffusion equations that is consistent with the discretization of the
transport equation is usually considered a sufficient condition for a DSA method to be unconditionally
effective [1-3]. However, the degradation of DSA methods - even fully consistent ones - in problems with
discontinuities in material properties means that consistency is not enough to guarantee the effectiveness of
a DSA method. This was first identified in  and  and revealed as a general deficiency of DSA in a
paper at this conference.
For this paper, we follow on the work of Ashby, et al. , Brown , and Guthrie, et al. , where Krylov
methods preconditioned by DSA replace traditional source iteration on the scalar flux, and extend their
approach to our linear discontinuous finite element method (DFEM) on unstructured tetrahedral grids. This
discretization, including a discussion of compatible DSA methods, is presented in Sec. 2.
We find that using a more powerful iteration, like a Krylov subspace iterative method , significantly
improves convergence for problems in which the convergence of accelerated source iteration degraded in
the presence of material discontinuities. A nice feature is that the Krylov iterative method can be
"wrapped around" source iteration code so that only minor changes to the original inner iteration coding is
necessary. A brief discussion of the formulation and implementation of the preconditioned Krylov iterative
solution method, including an overview of related work, is presented in Sec. 3
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Warsa, J. S. (James S.); Wareing, T. A. (Todd A.) & Morel, J. E. Krylov iterative methods applied to multidimensional S[sub n] calculations in the presence of material discontinuities, article, January 1, 2002; United States. (digital.library.unt.edu/ark:/67531/metadc934055/m1/2/: accessed January 22, 2019), University of North Texas Libraries, Digital Library, digital.library.unt.edu; crediting UNT Libraries Government Documents Department.