Optimal prediction for radiative transfer: A new perspective on moment closure

TL;DR

This paper introduces a novel perspective on moment closures in radiative transfer by applying optimal prediction, a method from nonlinear ODEs, to derive and analyze various existing and new closure schemes.

Contribution

It generalizes optimal prediction to PDE systems and re-derives classical linear closures, offering new insights and potential modifications for moment closure methods.

Findings

Re-derivation of $P_N$, diffusion, and diffusion correction closures using optimal prediction.

Provides a new perspective on existing moment closures in radiative transfer.

Suggests ideas for improving and modifying current closure schemes.

Abstract

Moment methods are classical approaches that approximate the mesoscopic radiative transfer equation by a system of macroscopic moment equations. An expansion in the angular variables transforms the original equation into a system of infinitely many moments. The truncation of this infinite system is the moment closure problem. Many types of closures have been presented in the literature. In this note, we demonstrate that optimal prediction, an approach originally developed to approximate the mean solution of systems of nonlinear ordinary differential equations, can be used to derive moment closures. To that end, the formalism is generalized to systems of partial differential equations. Using Gaussian measures, existing linear closures can be re-derived, such as $P_N$, diffusion, and diffusion correction closures. This provides a new perspective on several approximations done in the process and gives rise to ideas for modifications to existing closures.