Subcell balance methods for radiative transfer on arbitrary grids

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Abstract
We present a new spatial discretization method, which enforces conservation on quadrilateral subcells in an arbitrarily connected grid of polygonal cells, for two-dimensional radiative transfer problems. We review what is known about the performance of existing methods for optically thick, diffusive regions of radiative transfer problems, focusing in particular on bilinear discontinuous (BLD) finite-element methods and the simple corner-balance (SCB) method. We discuss the close relation of the SCB and BLD methods, and how they differ. By careful analysis, we relate specific properties of the SCB solution to specific approximations in the SCB method. We then build our new method by discarding those SCB approximations that lead to undesirable properties and carefully constructing new approximations designed to yield more desirable properties. We compare BLD, SCB. and the new scheme on a series of test problems in slab and XY geometries; numerical results invariably agree with predictions of the analysis. The new method matches SCB in thick diffusive regions, as it was designed to, but vastly outperforms SCB given cells of fine and intermediate optical thickness, where it achieves results comparable to BLD.