# The equilibrium-diffusion limit for radiation hydrodynamics

## TL;DR

This paper derives and analyzes the equilibrium-diffusion approximation (EDA) for radiation hydrodynamics, demonstrating its first-order accuracy and how common approximations preserve this accuracy in different frames, with practical implications for radiative-shock solutions.

## Contribution

The paper provides a rigorous asymptotic derivation of the EDA and shows that it and related approximations maintain first-order accuracy in both comoving and lab frames.

## Key findings

- EDA is first-order accurate with second-order transport corrections.
- Grey nonequilibrium-diffusion and Eddington approximations preserve first-order accuracy.
- Radiative-shock solutions approach equilibrium-diffusion solutions as the asymptotic parameter diminishes.

## Abstract

The equilibrium-diffusion approximation (EDA) is used to describe certain radiation-hydrodynamic (RH) environments. When this is done the RH equations reduce to a simplified set of equations. The EDA can be derived by asymptotically analyzing the full set of RH equations in the equilibrium-diffusion limit. We derive the EDA this way and show that it and the associated set of simplified equations are both first-order accurate with transport corrections occurring at second order. Having established the EDA's first-order accuracy we then analyze the grey nonequilibrium-diffusion approximation and the grey Eddington approximation and show that they both preserve this first-order accuracy. Further, these approximations preserve the EDA's first-order accuracy when made in either the comoving-frame (CMF) or the lab-frame (LF). While analyzing the Eddington approximation, we found that the CMF and LF radiation-source equations are equivalent when neglecting ${\cal O}(\beta^2)$ terms and compared in the LF. Of course, the radiation pressures are not equivalent. It is expected that simplified physical models and numerical discretizations of the RH equations that do not preserve this first-order accuracy will not retain the correct equilibrium-diffusion solutions. As a practical example, we show that nonequilibrium-diffusion radiative-shock solutions devolve to equilibrium-diffusion solutions when the asymptotic parameter is small.