# Krylov iterative methods applied to multidimensional S[sub n] calculations in the presence of material discontinuities Page: 4 of 20

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Krylov Methods for SN Problems with Material Discontinuities

N

01,k = m n(f m m,k ,(2c)

M=1

for all linear trial functions uj, j = 1, ... , 4 on cell Tk. The Galerkin approximation takes the trial

functions to be the basis functions Lj, and the above expressions can be evaluated for each of these four

functions. This gives four equations for the four unknowns V/'m,j,k on the cell. Before carrying out the

integrations in (2b), however, we first introduce the discontinuous approximation. Considering a cell k with

face p whose outward normal is np, the boundary terms o are defined as

nm +ny (p),k, gym. -pJ4, 2 nj > O, np in V

(0m. -p)@ = A_ -' mv tp, Am- np < 0, hp in V \aV (2d)

~m- npy t(), m" -p < 0, np on V

where l is the cell that shares face p with cell k. The subscript i(p) denotes three vertices i on a face p of a

given cell. Simply put, if np is on the boundary of the problem domain V, then the boundary condition is

used to define the incoming angular flux for the three points on a face; otherwise the internal or external

values angular fluxes are used depending on the orientation of the cell face with respect to the quadrature

direction. The discrete boundary conditions are vacuum, F(Alm) = 0, or F(Am) = for reflective

boundary conditions, where m'. is determined by the relationship

Am' = Am - 2 n (fm - n) , (2e)

for Am and n = hp. In our application, reflection is implemented only for boundary faces aligned parallel

to the x, y or z coordinate axes so that the standard quadrature sets we use contain the reflected angles 171m'

that satisfy this relationship.

The integrals in (2) are evaluated, either analytically or by quadrature approximation, for every cell in the

mesh. The angular flux, ')m,j,k, can then be computed for all vertices j = 1, 4 of every cell k, one cell at a

time over the entire mesh in a predetermined order for every quadrature angle ntm. Note that we use a fully

lumped version of (2). Describing it goes beyond the scope of this work, but suffice it to say that this

lumping preserves the diffusion limit in thick, diffusive regimes (see [15]).

2.2 Source Iteration in Operator Notation

In this section, we formulate the transport equation in an operator notation to facilitate our presentation.

We assume that we are using a standard SN angular discretization. Postponing discussion of the boundary

conditions, the discretized transport problem reads

Lo = MSDb + q. (3)

We use an N-point quadrature and there N, spatial cells in the problem. Let 0 be the vector of angular

fluxes for every angle and every vertex (four of them) in each cell, so 0 is of length n = 4NNc. The vector

q is source vector also of length n. The (n x n) operator L represents the discretized streaming and

removal operator for all angles. The vector 0 contains the Nm (this number depends on L and the particular

quadrature) of scalar flux moments at the four vertices of each cell, so that is is of length t = 4NmN. The

operator D maps the vector 0 onto 0, 0 = DV. The operator M maps a vector of scalar flux momentsAmerican Nuclear Society Topical Meeting in Mathematics & Computations, Gatlinburg, TN, 2003

3/19

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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. (digital.library.unt.edu/ark:/67531/metadc934055/m1/4/: accessed January 22, 2019), University of North Texas Libraries, Digital Library, digital.library.unt.edu; crediting UNT Libraries Government Documents Department.