Moment Closures Based on Minimizing the Residual of the P$_N$ Angular Expansion in Radiation Transport

Abstract

In this work we present two new closures for the spherical harmonics (PN) method in slab geometry transport problems. Our approach begins with an analysis of the squared-residual of the transport equation where we show that the standard truncation and diffusive closures do not minimize the residual of the PN expansion. Based on this analysis we derive two models, a moment-limited diffusive MLDN closure and a transient PN (TPN) closure that attempt to address shortcomings of common closures. The form of these closures is similar to flux-limiters for diffusion with the addition of a time-derivative in the definition of the closure. Numerical results on a pulsed plane source problem, the Gordian knot of slab-geometry transport problems, indicate that our new closure outperforms existing linear closures. Additionally, on a deep penetration problem we demonstrate that the TPN closure does not suffer from the artificial shocks that can arise in the MN entropy-based closure. Finally, results for Reed's problem demonstrate that the TPN solution is as accurate as the PN+3 solution. We further extend the TPN closure to 2D Cartesian geometry. The line source test problem demonstrates the model effectively damps oscillations and negative densities