Application of the Karhunen–Loève Transform to the C5G7 benchmark in the response matrix method

Presented is an application of the Karhunen–Loève Transform (KLT) for treatment of the energy variable in response matrix methods to a 44-group version of the 2-D C5G7 benchmark problem. Response matrix methods are based on the partitioning of global domains into independent nodes linked via boundary conditions approximated by truncated expansions of the phase space in orthogonal bases. Here, the KLT was used to produce basis sets appropriate for the energy variable based on “snapshots.” The method of snapshots employs small, representative problems to provide input spectra with which the KLT produces an orthogonal basis for the application. For this study, several computationally small models were defined, and the success of the corresponding basis was compared to the reference (full-multigroup) solution. The best performing basis sets were generated using information from both the scalar flux and the partial current, and typically included information from each unique material (e.g., a UO₂ pin cell) in the application problem and junctions between such materials (e.g., a UO₂ adjacent to a MOX pin cell). In general, the KLT performed better than the standard discrete Legendre Polynomials (DLPs) as well as “modified” DLPs with proper snapshot selection. The largest errors were found at the material junctions, precisely where spectral gradients are greatest.