# Spatial discretizations for self-adjoint forms of the radiative transfer equations

## Abstract

There are three commonly recognized second-order self-adjoint forms of the neutron transport equation: the even-parity equations, the odd-parity equations, and the self-adjoint angular flux equations. Because all of these equations contain second-order spatial derivatives and are self-adjoint for the mono-energetic case, standard continuous finite-element discretization techniques have proved quite effective when applied to the spatial variables. We first derive analogs of these equations for the case of time-dependent radiative transfer. The primary unknowns for these equations are functions of the angular intensity rather than the angular flux, hence the analog of the self-adjoint angular flux equation is referred to as the self-adjoint angular intensity equation. Then we describe a general, arbitrary-order, continuous spatial finite-element approach that is applied to each of the three equations in conjunction with backward-Euler differencing in time. We refer to it as the 'standard' technique. We also introduce an alternative spatial discretization scheme for the self-adjoint angular intensity equation that requires far fewer unknowns than the standard method, but appears to give comparable accuracy. Computational results are given that demonstrate the validity of both of these discretization schemes.

- Authors:

- Texas A and M University, Department of Nuclear Engineering, 129 Zachry Engineering Center, College Station, TX 77843-3133 (United States). E-mail: morel@tamu.edu
- Computer and Computational Sciences Division, Los Alamos National Laboratory, Los Alamos, NM 87545 (United States)
- (Korea, Republic of)
- Transpire Technologies, 6659 Kimball Drive, Suite D-404, Gig Harbor, Washington 98335 (United States)

- Publication Date:

- OSTI Identifier:
- 20767039

- Resource Type:
- Journal Article

- Resource Relation:
- Journal Name: Journal of Computational Physics; Journal Volume: 214; Journal Issue: 1; Other Information: DOI: 10.1016/j.jcp.2005.09.017; PII: S0021-9991(05)00421-3; Copyright (c) 2005 Elsevier Science B.V., Amsterdam, The Netherlands, All rights reserved; Country of input: International Atomic Energy Agency (IAEA)

- Country of Publication:
- United States

- Language:
- English

- Subject:
- 71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSICS; ACCURACY; EQUATIONS; FINITE ELEMENT METHOD; NEUTRON TRANSPORT THEORY; RADIANT HEAT TRANSFER; THERMAL RADIATION; TIME DEPENDENCE

### Citation Formats

```
Morel, Jim E., Adams, B. Todd, Noh, Taewan, Department of Mathematics, Hongik University, Seoul, McGhee, John M., Evans, Thomas M., and Urbatsch, Todd J..
```*Spatial discretizations for self-adjoint forms of the radiative transfer equations*. United States: N. p., 2006.
Web. doi:10.1016/j.jcp.2005.09.017.

```
Morel, Jim E., Adams, B. Todd, Noh, Taewan, Department of Mathematics, Hongik University, Seoul, McGhee, John M., Evans, Thomas M., & Urbatsch, Todd J..
```*Spatial discretizations for self-adjoint forms of the radiative transfer equations*. United States. doi:10.1016/j.jcp.2005.09.017.

```
Morel, Jim E., Adams, B. Todd, Noh, Taewan, Department of Mathematics, Hongik University, Seoul, McGhee, John M., Evans, Thomas M., and Urbatsch, Todd J.. Mon .
"Spatial discretizations for self-adjoint forms of the radiative transfer equations". United States.
doi:10.1016/j.jcp.2005.09.017.
```

```
@article{osti_20767039,
```

title = {Spatial discretizations for self-adjoint forms of the radiative transfer equations},

author = {Morel, Jim E. and Adams, B. Todd and Noh, Taewan and Department of Mathematics, Hongik University, Seoul and McGhee, John M. and Evans, Thomas M. and Urbatsch, Todd J.},

abstractNote = {There are three commonly recognized second-order self-adjoint forms of the neutron transport equation: the even-parity equations, the odd-parity equations, and the self-adjoint angular flux equations. Because all of these equations contain second-order spatial derivatives and are self-adjoint for the mono-energetic case, standard continuous finite-element discretization techniques have proved quite effective when applied to the spatial variables. We first derive analogs of these equations for the case of time-dependent radiative transfer. The primary unknowns for these equations are functions of the angular intensity rather than the angular flux, hence the analog of the self-adjoint angular flux equation is referred to as the self-adjoint angular intensity equation. Then we describe a general, arbitrary-order, continuous spatial finite-element approach that is applied to each of the three equations in conjunction with backward-Euler differencing in time. We refer to it as the 'standard' technique. We also introduce an alternative spatial discretization scheme for the self-adjoint angular intensity equation that requires far fewer unknowns than the standard method, but appears to give comparable accuracy. Computational results are given that demonstrate the validity of both of these discretization schemes.},

doi = {10.1016/j.jcp.2005.09.017},

journal = {Journal of Computational Physics},

number = 1,

volume = 214,

place = {United States},

year = {Mon May 01 00:00:00 EDT 2006},

month = {Mon May 01 00:00:00 EDT 2006}

}