Adaptive characteristic spatial quadratures for discrete ordinates neutral particle transport—the rectangular cell case

Abstract

Abstract Adaptive characteristic spatial quadrature schemes for use in discrete ordinates neutral particle transport computer codes in slab geometry have been introduced recently; the extension to two-dimensional Cartesian coordinates with rectangular cells is presented here. These quadratures differ from previous schemes in using strictly non-negative finite-element-like representations of the entering fluxes and scattering source, which adapt to match both the zeroth and first spatial moments. (These input moments are obtained from the boundary conditions or as output moments of the upstream adjacent cells and from the angular quadrature of the spatial flux moments in the outer iteration, respectively.) The flux throughout the cell is found by integration of the transport equation along streaming (characteristic) lines. The moments of the flux through the cell and along the exiting boundaries are then obtained. The resulting quadratures are nonlinear and strictly non-negative, while satisfying zeroth-spatial-moment and first-spatial-moment balances within each cell and globally. For a square test problem 32 mean free paths thick, the variation in results over a range of meshes from 1 by 1 to 8 by 8 was comparable to (usually less than) the corresponding variation due to angular quadrature (S4, S8, S16). Linear adaptive quadrature performed better than step adaptive, with few exceptions. Using a mesh consisting of a single cell, both adaptive methods performed as well as or better than linear nodal and linear characteristic methods did with an 8 by 8 mesh of cells. Although they are substantially more complicated to derive and implement, the adaptive methods cost only three to nine times more per cell to compute, and require no additional storage, as compared to linear nodal and linear characteristic methods. Thus, by using these coarser meshes, step and linear adaptive were more cost-effective than linear nodal or linear characteristic (with or without negative fixups). Linear adaptive is similar to linear characteristic (but with a bilinear term) wherever linear characteristic requires no negative flux or source fixups. Thus, it should share the excellent convergence of linear characteristic. For an optically thin test problem, with a fine mesh (up to 32 by 32 cell mesh on a square 2 mfp thick), we found that linear adaptive was slightly more accurate than linear characteristic without fixup, which was substantially different than linear characteristic with fixup. This test showed that, even with a fine mesh, fixups can be required by conventional methods (linear nodal and linear characteristic), which either have negative fluxes or violate first-spatial-moment balance. Linear adaptive has no negative fluxes and conserves first spatial moments; its results were very close to those of linear characteristic (without fixup), for optically thin cells. Step adaptive converges somewhat more slowly, however; so it is less efficient than the other methods when both are used on a fine enough mesh. We conclude that linear adaptive quadrature can be an accurate and cost-effective method for problems requiring the use of optically thick cells.