# The equilibrium-diffusion limit for radiation hydrodynamics

## 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. Here, 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). And while analyzing the Eddington approximation, we found that the CMF and LF radiation-source equations are equivalent when neglecting O(β ^{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 parametermore »

- Authors:

- Los Alamos National Lab. (LANL), Los Alamos, NM (United States)
- Texas A & M Univ., College Station, TX (United States)

- Publication Date:

- Research Org.:
- Los Alamos National Lab. (LANL), Los Alamos, NM (United States)

- Sponsoring Org.:
- USDOE

- OSTI Identifier:
- 1374348

- Report Number(s):
- LA-UR-17-20878

Journal ID: ISSN 0022-4073

- Grant/Contract Number:
- AC52-06NA25396

- Resource Type:
- Journal Article: Accepted Manuscript

- Journal Name:
- Journal of Quantitative Spectroscopy and Radiative Transfer

- Additional Journal Information:
- Journal Volume: 202; Journal Issue: C; Journal ID: ISSN 0022-4073

- Publisher:
- Elsevier

- Country of Publication:
- United States

- Language:
- English

- Subject:
- 73 NUCLEAR PHYSICS AND RADIATION PHYSICS; Asymptotics; Equilibrium diffusion; Radiation transport; Radiation hydrodynamics; Grey nonequilibrium-diffusion approximation; Grey Eddington approximation; Radiative-shock solutions

### Citation Formats

```
Ferguson, J. M., Morel, J. E., and Lowrie, R.
```*The equilibrium-diffusion limit for radiation hydrodynamics*. United States: N. p., 2017.
Web. doi:10.1016/j.jqsrt.2017.07.031.

```
Ferguson, J. M., Morel, J. E., & Lowrie, R.
```*The equilibrium-diffusion limit for radiation hydrodynamics*. United States. doi:10.1016/j.jqsrt.2017.07.031.

```
Ferguson, J. M., Morel, J. E., and Lowrie, R. Thu .
"The equilibrium-diffusion limit for radiation hydrodynamics". United States.
doi:10.1016/j.jqsrt.2017.07.031. https://www.osti.gov/servlets/purl/1374348.
```

```
@article{osti_1374348,
```

title = {The equilibrium-diffusion limit for radiation hydrodynamics},

author = {Ferguson, J. M. and Morel, J. E. and Lowrie, R.},

abstractNote = {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. Here, 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). And while analyzing the Eddington approximation, we found that the CMF and LF radiation-source equations are equivalent when neglecting O(β2) 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.},

doi = {10.1016/j.jqsrt.2017.07.031},

journal = {Journal of Quantitative Spectroscopy and Radiative Transfer},

number = C,

volume = 202,

place = {United States},

year = {Thu Jul 27 00:00:00 EDT 2017},

month = {Thu Jul 27 00:00:00 EDT 2017}

}