A review of the upstream corner balance spatial discretization

In this work, we document and study two variants of upstream corner balance (UCB) spatial discretizations for the linear Boltzmann equation. The first variant of UCB was published in 1997 by Adams. The second variant corrects a flaw, discovered in 2005, inherent in the 1997 variant. The second (2005) variant has not previously been published. Computational results are presented to: demonstrate the equivalence of the two UCB methods in slab geometry, verify that the second variant of UCB corrects the multi-dimensional flaw of the original UCB, and explore the accuracy of UCB relative to the more widely known and used bilinear discontinuous finite element spatial discretizations. Each UCB method studied here performs well in some limits but poorly in other limits. (authors)