An energy basis for response matrix methods based on the Karhunen-Loéve Transform

A new energy expansion technique based on the Karhunen-Loéve Transform (KLT) is developed for use in the eigenvalue response matrix method (ERMM). ERMM is a spatial domain decomposition method that links nodes using truncated expansions of boundary fluxes in each phase-space variable. Energy bases constructed using KLT can capture a comparatively large amount of spectral information in the first several basis functions, thus permitting low-order expansions with less error than expansions based on the more traditional Discrete Legendre Polynomials (DLP) or modified DLP's. The KLT basis functions are generated from representative energy spectra (called snapshots) for either the entire core model or various smaller models (called snapshot models) representing core components, e.g., pins or assemblies. Energy bases using KLT are compared to alternative bases using two test problems in either a 44-group or 238-group format. The results indicate that the performance of the KLT bases is not strongly dependent on the number of groups, and, hence, many-group fidelity can be captured in the first few basis functions. Using snapshots from the full model of interest to generate the basis functions can yield sub-0.1% relative error in the pin fission density in less than 10 energy degrees of freedom with a 238-group library. Using more practical snapshot models, e.g., assembly models for a full core problem, the same error can be reached with as few as 15 energy degrees of freedom.