| Deterministic radiation transport discretizations mimic, to various degrees, properties of the analytic transport equation. No deterministic scheme accurately mimics all of these properties, and one must generally choose a method that prioritizes, based on properties important to the problem at hand. When the problem contains optically-thick diffusive regions, for example, the importance of meeting the thick diffusion limit is well known.; We demonstrate that linear-solution-preservation (in space and angle) is also vitally important to obtaining accurate solutions in diffusive regions. Though this quality is achieved by modern Corner Balance/Discontinuous Finite Element Methods on triangular/tetrahedral grids, the state-of-the-art Upstream Corner Balance method fails to preserve linear-solutions on quadrilateral grids. We develop a new Multidimensional Multiple Balance method, which is linear-solution-preserving on skewed quadrilateral grids. We demonstrate the new method's success in diffusive regions on non-orthogonal quadrilateral grids, and we compare it to Upstream Corner Balance on a variety of other problems. Finally, we develop two physics-based preconditioners for the new method on orthogonal grids. |
