A Generalized Multigroup Method

The standard multigroup (MG) method for energy discretization of the transport equation can be sensitive to approximation in the weighting spectrum chosen for cross-section averaging. As a result, MG often inaccurately treats important phenomena such as self-shielding variations across a fuel pin in nuclear reactor simulations. From a finite-element viewpoint, MG uses a single fixed basis function (the pre-selected spectrum) within each group, with no mechanism to adapt to local spatial and angular solution realities. To address these issues, we introduce a Petrov-Galerkin finite-element multigroup (PG-FEMG) method, a generalization of the MG method that is related to the family of multiband (MB) methods. PG-FEMG uses integrals over several discontinuous energy domains within a group as its degrees of freedom, which allows PG-FEMG to be used in standard MG-based computer codes with changes to pre- and post-processing of the data only. We define a problem-wide effective total cross section as the basis of these discontinuous energy domains.

We implement the PG-FEMG method for several realistic pin-cell problems and find it to be significantly more accurate per degree of freedom than MG for several quantities of interest, including criticality eigenvalue and power profile shape. We find that PG-FEMG is much less sensitive to errors in weighting spectra compared to standard MG. We discuss straightforward generalizations to multi-dimensional problems of practical interest, including reactor depletion calculations.