A characteristic-based multiple balance approach for solving the S(N) equations of arbitrary polygonal meshes

We introduce a new approach for solving the neutral particle transport problem on arbitrary two-dimensional meshes in a multigroup discrete ordinates (S{dollar}rmsb{lcub}n{rcub}{dollar}) context. Our approach includes spatial discretization of regions with uniform material properties and solution of the transport problem with prescribed conditions. The spatial discretization involves approximating a general surface as an arbitrary polygon, rotating to a coordinate system aligned with the direction of particle travel, and decomposing the polygonal cell into subregions called slices. This simple and intuitively appealing geometry decomposition follows from a characteristic-based view of the transport problem. Most balance-based characteristic methods use this decomposition implicitly: we include it explicitly and exploit its properties. Our mathematical approach to the transport problem is a multiple balance approach, using exact spatial moments balance equations on whole cells and slices, together with approximate auxiliary equations on slices to close the system. Therefore, we call our approach the slice balance approach and describe it as a characteristic-based multiple balance approach. In our view, the most salient result of this research is that our slice balance approach is very general and facilitates extension of planar geometry S{dollar}rmsb{lcub}n{rcub}{dollar} spatial differencing schemes to arbitrarily complex polygonal meshes.; We derive a general-order characteristic family of slice balance schemes, along with specific step- and linear-characteristic schemes. We also derive a simple "diamond-difference-like" scheme to demonstrate the flexibility of the slice balance approach. The step characteristic scheme is implemented in the standard computational module of the CENTAUR-123 computer code package. The other schemes are implemented in new proof-of-principle computational modules. Our current implementations are limited to isotropic scattering and vacuum boundary conditions, but extensions to general scattering and boundary conditions are straightforward. Numerical results are compared against analytical solutions and against industry-accepted S{dollar}rmsb{lcub}n{rcub}{dollar} computer codes for a range of problems. These results demonstrate the ability of the slice balance approach to closely approximate complicated geometric configurations. Properties of the numerical solutions depend on the differencing scheme used within the slice balance framework and are analogous to properties observed in traditional methods based on the same differencing.