Optimal prediction for moment models: Crescendo diffusion and reordered equations

TL;DR

This paper introduces a unified framework for moment closures in radiative transfer using optimal prediction, deriving existing closures and proposing new ones like crescendo diffusion, which improves approximation accuracy.

Contribution

It applies optimal prediction to derive and unify various moment closures, introduces reordered $P_N$ equations, and proposes a novel crescendo diffusion closure with enhanced accuracy.

Findings

Re-derivation of $P_N$, diffusion, and diffusion correction closures within the framework.

Introduction of reordered $P_N$ equations similar to simplified $P_N$.

Crescendo diffusion closure improves approximation accuracy in numerical tests.

Abstract

A direct numerical solution of the radiative transfer equation or any kinetic equation is typically expensive, since the radiative intensity depends on time, space and direction. An expansion in the direction variables yields an equivalent system of infinitely many moments. A fundamental problem is how to truncate the system. Various closures have been presented in the literature. We want to study moment closure generally within the framework of optimal prediction, a strategy to approximate the mean solution of a large system by a smaller system, for radiation moment systems. We apply this strategy to radiative transfer and show that several closures can be re-derived within this framework, e.g. $P_N$, diffusion, and diffusion correction closures. In addition, the formalism gives rise to new parabolic systems, the reordered $P_N$ equations, that are similar to the simplified $P_N$ equations. Furthermore, we propose a modification to existing closures. Although simple and with no extra cost, this newly derived crescendo diffusion yields better approximations in numerical tests.