## A High-Order Low-Order Algorithm with Exponentially Convergent Monte Carlo for Thermal Radiative Transfer

## Abstract

In this paper, we have implemented a new high-order low-order (HOLO) algorithm for solving thermal radiative transfer problems. The low-order (LO) system is based on the spatial and angular moments of the transport equation and a linear-discontinuous finite-element spatial representation, producing equations similar to the standard S _{2} equations. The LO solver is fully implicit in time and efficiently resolves the nonlinear temperature dependence at each time step. The high-order (HO) solver utilizes exponentially convergent Monte Carlo (ECMC) to give a globally accurate solution for the angular intensity to a fixed-source pure-absorber transport problem. This global solution is used to compute consistency terms, which require the HO and LO solutions to converge toward the same solution. The use of ECMC allows for the efficient reduction of statistical noise in the Monte Carlo solution, reducing inaccuracies introduced through the LO consistency terms. Finally, we compare results with an implicit Monte Carlo code for one-dimensional gray test problems and demonstrate the efficiency of ECMC over standard Monte Carlo in this HOLO algorithm.

- Authors:

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

- Publication Date:

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

- Sponsoring Org.:
- USDOE Office of Nuclear Energy (NE). Nuclear Energy University Programs (NEUP); USDOE National Nuclear Security Administration (NNSA)

- OSTI Identifier:
- 1338753

- Report Number(s):
- LA-UR-16-20497

Journal ID: ISSN 0029-5639; TRN: US1701348

- Grant/Contract Number:
- AC52-06NA25396; NA0002376

- Resource Type:
- Accepted Manuscript

- Journal Name:
- Nuclear Science and Engineering

- Additional Journal Information:
- Journal Volume: 185; Journal Issue: 1; Journal ID: ISSN 0029-5639

- Publisher:
- American Nuclear Society

- Country of Publication:
- United States

- Language:
- English

- Subject:
- 97 MATHEMATICS AND COMPUTING; Thermal Radiative Transfer, HOLO, IMC, Monte Carlo

### Citation Formats

```
Bolding, Simon R., Cleveland, Mathew Allen, and Morel, Jim E. A High-Order Low-Order Algorithm with Exponentially Convergent Monte Carlo for Thermal Radiative Transfer. United States: N. p., 2016.
Web. doi:10.13182/NSE16-36.
```

```
Bolding, Simon R., Cleveland, Mathew Allen, & Morel, Jim E. A High-Order Low-Order Algorithm with Exponentially Convergent Monte Carlo for Thermal Radiative Transfer. United States. doi:10.13182/NSE16-36.
```

```
Bolding, Simon R., Cleveland, Mathew Allen, and Morel, Jim E. Fri .
"A High-Order Low-Order Algorithm with Exponentially Convergent Monte Carlo for Thermal Radiative Transfer". United States. doi:10.13182/NSE16-36. https://www.osti.gov/servlets/purl/1338753.
```

```
@article{osti_1338753,
```

title = {A High-Order Low-Order Algorithm with Exponentially Convergent Monte Carlo for Thermal Radiative Transfer},

author = {Bolding, Simon R. and Cleveland, Mathew Allen and Morel, Jim E.},

abstractNote = {In this paper, we have implemented a new high-order low-order (HOLO) algorithm for solving thermal radiative transfer problems. The low-order (LO) system is based on the spatial and angular moments of the transport equation and a linear-discontinuous finite-element spatial representation, producing equations similar to the standard S2 equations. The LO solver is fully implicit in time and efficiently resolves the nonlinear temperature dependence at each time step. The high-order (HO) solver utilizes exponentially convergent Monte Carlo (ECMC) to give a globally accurate solution for the angular intensity to a fixed-source pure-absorber transport problem. This global solution is used to compute consistency terms, which require the HO and LO solutions to converge toward the same solution. The use of ECMC allows for the efficient reduction of statistical noise in the Monte Carlo solution, reducing inaccuracies introduced through the LO consistency terms. Finally, we compare results with an implicit Monte Carlo code for one-dimensional gray test problems and demonstrate the efficiency of ECMC over standard Monte Carlo in this HOLO algorithm.},

doi = {10.13182/NSE16-36},

journal = {Nuclear Science and Engineering},

number = 1,

volume = 185,

place = {United States},

year = {2016},

month = {10}

}