Nuclear Science and Engineering / Volume 162 / Number 3 / July 2009 / Pages 234-252
General expressions for the matrix elements of the discrete SN-equivalent integral transport operator are derived in slab geometry. Their asymptotic behavior versus cell optical thickness is investigated both for a homogeneous slab and for a heterogeneous slab characterized by a periodic material discontinuity wherein each optically thick cell is surrounded by two optically thin cells in a repeating pattern. In the case of a homogeneous slab, the asymptotic analysis conducted in the thick-cell limit for a highly scattering medium shows that the discretized integral transport operator approaches a tridiagonal matrix possessing a diffusion-like coupling stencil. It is further shown that this structure is approached at a fast exponential rate with increasing cell thickness when the arbitrarily high order transport method of the nodal type and zero-order spatial approximation (AHOT-N0) formalism is employed to effect the spatial discretization of the discrete ordinates transport operator. In the case of periodically heterogeneous slab configurations, the asymptotic behavior is realized by pushing apart the cells' optical thicknesses; i.e., the thick cells are made thicker while the thin cells are made thinner at a prescribed rate. We show that in this limit the discretized integral transport operator is approximated by a pentadiagonal structure. Notwithstanding, the discrete operator is amenable to algebraic transformations leading to a matrix representation still asymptotically approaching a tridiagonal structure at a fast exponential rate bearing close resemblance to the diffusive operator.
The results of the asymptotic analysis of the integral transport matrix are then used to gain insight into the excellent convergence properties of the adjacent-cell preconditioner (AP) acceleration scheme. Specifically, the AP operator exactly captures the asymptotic structure acquired by the integral transport matrix in the thick-cell limit for homogeneous slabs of pure-scatterer or partial-scatterer material, and for periodically heterogeneous slabs hosting purely scattering materials. In the above limits the integral transport matrix reduces to a diffusive structure consistent with the diffusive matrix template used to construct the AP. In the case of periodically heterogeneous slabs containing absorbing materials, the AP operator partially captures the asymptotic structure acquired by the integral transport matrix. The inexact agreement is due either to discrepancies in the equations for the boundary cells or to the nondiffusive structure acquired by the integral transport matrix. These findings shed light on the immediate convergence, i.e., convergence in two iterations, displayed by the AP acceleration scheme in the asymptotic limit for slabs hosting purely scattering materials, both in the homogeneous and periodically heterogeneous cases. For periodically heterogeneous slabs containing absorbing materials, immediate convergence is achieved by modifying the original recipe for constructing the AP so that the correct asymptotic structure of the integral transport matrix coincides with the AP operator in the asymptotic limit.