Krylov iterative methods applied to multidimensional S[sub n] calculations in the presence of material discontinuities Page: 3 of 20
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Warsa, Wareing, and Morel
We show numerical results for a realistic, unstructured mesh problem with a range of differing material
properties in Sec. 4. We find that (a) the Krylov method alone, without DSA preconditioning, accelerates
the transport solution, although not always as well as accelerated source iteration, (b) convergence is vastly
improved if the Krylov method is preconditioned with DSA, and (c) the Krylov method restores the
effectiveness of the partially consistent, but inexpensive, simplified WLA (S-WLA) DSA scheme [10, 11].
This last observation is also true for the fully consistent DSA method, but the high costs we encountered in
the fully consistent equations is unacceptable [12, 13]. It is particularly costly in problems with material
discontinuities. Therefore we only consider the less costly S-WLA method for preconditioning the Krylov
method.
2 DISCONTINUOUS FINITE ELEMENT DISCRETIZATION ON TETRAHEDRAL MESHES
We present the linear discontinuous finite element method (DFEM) for the SN transport equation on
unstructured tetrahedral meshes, followed by a brief overview of DSA. Further details on the fully
consistent DSA scheme can be found in [13] and details of the partially consistent DSA method can be
found in [10] and [11].
2.1 Discontinuous Finite Element Discretization
The notation used here has the usual meaning [14] and we assume cgs units. Given an angular quadrature
set with N specified nodes and weights {m, wm}, a distributed source of particles Q(r, !l) and
anisotropic scattering of order L, the monoenergetic, steady-state SN transport equation in the
three-dimensional domain r E V with boundary r, E V, is
L l
Alm.VPm(r) + at(r)m(r) = 0s,l Yin(f m)q$ (r) + Q(r, Oim), m = 1, . . , N. (la)
l=0 n=-l
Here, Yin(0) are the normalized spherical harmonics functions and the scalar flux moments are
#f (r) = wmYin(A2m)m(r). (lb)
m=1
The inhomogeneous source is assumed to be isotropic, or Q(r, f2) = Qo(r).
The linear DFEM discretization is specified by the following variational formulation. It is written in source
iteration form with iteration index e. Given an angular flux expansion in terms of the four independent
linear basis functions on a tetrahedral cell Tk,
4
m,k m,j,kLj(r), (2a)
j=1
find the linear approximation for each angle Am that satisfies
\ k kkmTk / Jk
L l (2b)
= Os,l,k Yn(Qm) J Ik uj dV + Qouk dV ,
l=0 n=-l kkAmerican Nuclear Society Topical Meeting in Mathematics & Computations, Gatlinburg, TN, 2003
2/19
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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. (https://digital.library.unt.edu/ark:/67531/metadc934055/m1/3/: accessed April 24, 2024), University of North Texas Libraries, UNT Digital Library, https://digital.library.unt.edu; crediting UNT Libraries Government Documents Department.