A positive non-linear closure for the SN equations with linear-discontinous spatial differencing
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We have developed a new parametric non-linear closure for the 1-D slab-geometry Sn equations with linear-discontinuous (LD) spatial differencing that is strictly positive and yields the set-to-zero fixup equations in the limit as the parameter is increased without bound. Unlike the standard LD equations with set-to-zero fixup, these non-linear Sn equations, for any finite value of the parameter, are differentiable and thus amenable to solution via Newton's method. Furthermore, unlike any exponential-based closure method, our new scheme is robust with respect to negativities in the scattering source that often arise with highly anisotropic scattering. We present results indicating that for an appropriate range of parameteric values, our new method is strictly positive, efficient, and yields solutions that rapidly approach the standard LD solution as the spatial mesh is refined.
