Second Order Closures for the Radiative Transfer Equation: Some Are Unstable

TL;DR

This paper investigates higher-order closure relations for the radiative transfer equation in cosmic reionization simulations, revealing that most natural generalizations are physically unstable, thus constraining future modeling approaches.

Contribution

It demonstrates that second-order closures beyond the first order are generally unstable, providing critical insights into the limitations of current radiative transfer modeling methods.

Findings

Higher-order closures like OTVET are unstable.

Local second-order closures depending only on intensity and flux are unstable.

Most natural generalizations of M1 and OTVET are physically unstable.

Abstract

The largest existing simulations of cosmic reionization model radiative transfer with moment methods that require a closure relation. The two most commonly used closure relations are M1 and OTVET; both close the moment hierarchy at the first moment. We explore the properties of a higher, second-order closure. We show that direct generalizations of M1 and OTVET to one higher order are physically unstable - i.e., the closure equations themselves result in unstable solutions, not just their numerical implementation. In fact, a generalization of OTVET to any order higher than the first one is unstable. We are also able to show that any local (i.e., depending only on the local moments of the radiation field, like M1) second-order closure that depends only on the radiation intensity and radiation flux, but does not explicitly depend on the radiation pressure, is physically unstable. This result restricts the choice of possible second-order closure relations.