# Accelerated solution of discrete ordinates approximation to the Boltzmann transport equation via model reduction

## Abstract

Radiation heat transfer is an important phenomenon in many physical systems of practical interest. When participating media is important, the radiative transfer equation (RTE) must be solved for the radiative intensity as a function of location, time, direction, and wavelength. In many heat-transfer applications, a quasi-steady assumption is valid, thereby removing time dependence. The dependence on wavelength is often treated through a weighted sum of gray gases (WSGG) approach. The discrete ordinates method (DOM) is one of the most common methods for approximating the angular (i.e., directional) dependence. The DOM exactly solves for the radiative intensity for a finite number of discrete ordinate directions and computes approximations to integrals over the angular space using a quadrature rule; the chosen ordinate directions correspond to the nodes of this quadrature rule. This paper applies a projection-based model-reduction approach to make high-order quadrature computationally feasible for the DOM for purely absorbing applications. First, the proposed approach constructs a reduced basis from (high-fidelity) solutions of the radiative intensity computed at a relatively small number of ordinate directions. Then, the method computes inexpensive approximations of the radiative intensity at the (remaining) quadrature points of a high-order quadrature using a reduced-order model constructed from the reducedmore »

- Authors:

- Sandia National Lab. (SNL-NM), Albuquerque, NM (United States)
- Sandia National Lab. (SNL-CA), Livermore, CA (United States)

- Publication Date:

- Research Org.:
- Sandia National Lab. (SNL-NM), Albuquerque, NM (United States)

- Sponsoring Org.:
- USDOE Laboratory Directed Research and Development (LDRD) Program

- OSTI Identifier:
- 1372351

- Report Number(s):
- SAND-2017-1114J

Journal ID: ISSN 0022-1481; 650925

- Grant/Contract Number:
- AC04-94AL85000

- Resource Type:
- Journal Article: Accepted Manuscript

- Journal Name:
- Journal of Heat Transfer

- Additional Journal Information:
- Journal Volume: 139; Journal Issue: 12; Journal ID: ISSN 0022-1481

- Publisher:
- ASME

- Country of Publication:
- United States

- Language:
- English

- Subject:
- 73 NUCLEAR PHYSICS AND RADIATION PHYSICS; approximation; heat transfer; wavelength; gases; radiation (physics); radiative heat transfer

### Citation Formats

```
Tencer, John, Carlberg, Kevin, Larsen, Marvin, and Hogan, Roy E.
```*Accelerated solution of discrete ordinates approximation to the Boltzmann transport equation via model reduction*. United States: N. p., 2017.
Web. doi:10.1115/1.4037098.

```
Tencer, John, Carlberg, Kevin, Larsen, Marvin, & Hogan, Roy E.
```*Accelerated solution of discrete ordinates approximation to the Boltzmann transport equation via model reduction*. United States. doi:10.1115/1.4037098.

```
Tencer, John, Carlberg, Kevin, Larsen, Marvin, and Hogan, Roy E. Sat .
"Accelerated solution of discrete ordinates approximation to the Boltzmann transport equation via model reduction". United States.
doi:10.1115/1.4037098. https://www.osti.gov/servlets/purl/1372351.
```

```
@article{osti_1372351,
```

title = {Accelerated solution of discrete ordinates approximation to the Boltzmann transport equation via model reduction},

author = {Tencer, John and Carlberg, Kevin and Larsen, Marvin and Hogan, Roy E.},

abstractNote = {Radiation heat transfer is an important phenomenon in many physical systems of practical interest. When participating media is important, the radiative transfer equation (RTE) must be solved for the radiative intensity as a function of location, time, direction, and wavelength. In many heat-transfer applications, a quasi-steady assumption is valid, thereby removing time dependence. The dependence on wavelength is often treated through a weighted sum of gray gases (WSGG) approach. The discrete ordinates method (DOM) is one of the most common methods for approximating the angular (i.e., directional) dependence. The DOM exactly solves for the radiative intensity for a finite number of discrete ordinate directions and computes approximations to integrals over the angular space using a quadrature rule; the chosen ordinate directions correspond to the nodes of this quadrature rule. This paper applies a projection-based model-reduction approach to make high-order quadrature computationally feasible for the DOM for purely absorbing applications. First, the proposed approach constructs a reduced basis from (high-fidelity) solutions of the radiative intensity computed at a relatively small number of ordinate directions. Then, the method computes inexpensive approximations of the radiative intensity at the (remaining) quadrature points of a high-order quadrature using a reduced-order model constructed from the reduced basis. Finally, this results in a much more accurate solution than might have been achieved using only the ordinate directions used to compute the reduced basis. One- and three-dimensional test problems highlight the efficiency of the proposed method.},

doi = {10.1115/1.4037098},

journal = {Journal of Heat Transfer},

number = 12,

volume = 139,

place = {United States},

year = {Sat Jun 17 00:00:00 EDT 2017},

month = {Sat Jun 17 00:00:00 EDT 2017}

}