# An asymptotic-preserving stochastic Galerkin method for the radiative heat transfer equations with random inputs and diffusive scalings

## Abstract

In this paper, we develop an Asymptotic-Preserving (AP) stochastic Galerkin scheme for the radiative heat transfer equations with random inputs and diffusive scalings. In this problem the random inputs arise due to uncertainties in cross section, initial data or boundary data. We use the generalized polynomial chaos based stochastic Galerkin (gPC-SG) method, which is combined with the micro–macro decomposition based deterministic AP framework in order to handle efficiently the diffusive regime. For linearized problem we prove the regularity of the solution in the random space and consequently the spectral accuracy of the gPC-SG method. We also prove the uniform (in the mean free path) linear stability for the space-time discretizations. Several numerical tests are presented to show the efficiency and accuracy of proposed scheme, especially in the diffusive regime.

- Authors:

- Department of Mathematics, University of Wisconsin-Madison, Madison, WI 53706 (United States)
- (China)

- Publication Date:

- OSTI Identifier:
- 22622269

- Resource Type:
- Journal Article

- Resource Relation:
- Journal Name: Journal of Computational Physics; Journal Volume: 334; Other Information: Copyright (c) 2016 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:
- 75 CONDENSED MATTER PHYSICS, SUPERCONDUCTIVITY AND SUPERFLUIDITY; ACCURACY; ASYMPTOTIC SOLUTIONS; CHAOS THEORY; CROSS SECTIONS; DECOMPOSITION; DIFFUSION; EFFICIENCY; EQUATIONS; HEAT; HEAT TRANSFER; MEAN FREE PATH; POLYNOMIALS; RANDOMNESS; SPACE-TIME; STABILITY; STOCHASTIC PROCESSES

### Citation Formats

```
Jin, Shi, E-mail: sjin@wisc.edu, Institute of Natural Sciences, Department of Mathematics, MOE-LSEC and SHL-MAC, Shanghai Jiao Tong University, Shanghai 200240, and Lu, Hanqing, E-mail: hanqing@math.wisc.edu.
```*An asymptotic-preserving stochastic Galerkin method for the radiative heat transfer equations with random inputs and diffusive scalings*. United States: N. p., 2017.
Web. doi:10.1016/J.JCP.2016.12.033.

```
Jin, Shi, E-mail: sjin@wisc.edu, Institute of Natural Sciences, Department of Mathematics, MOE-LSEC and SHL-MAC, Shanghai Jiao Tong University, Shanghai 200240, & Lu, Hanqing, E-mail: hanqing@math.wisc.edu.
```*An asymptotic-preserving stochastic Galerkin method for the radiative heat transfer equations with random inputs and diffusive scalings*. United States. doi:10.1016/J.JCP.2016.12.033.

```
Jin, Shi, E-mail: sjin@wisc.edu, Institute of Natural Sciences, Department of Mathematics, MOE-LSEC and SHL-MAC, Shanghai Jiao Tong University, Shanghai 200240, and Lu, Hanqing, E-mail: hanqing@math.wisc.edu. Sat .
"An asymptotic-preserving stochastic Galerkin method for the radiative heat transfer equations with random inputs and diffusive scalings". United States.
doi:10.1016/J.JCP.2016.12.033.
```

```
@article{osti_22622269,
```

title = {An asymptotic-preserving stochastic Galerkin method for the radiative heat transfer equations with random inputs and diffusive scalings},

author = {Jin, Shi, E-mail: sjin@wisc.edu and Institute of Natural Sciences, Department of Mathematics, MOE-LSEC and SHL-MAC, Shanghai Jiao Tong University, Shanghai 200240 and Lu, Hanqing, E-mail: hanqing@math.wisc.edu},

abstractNote = {In this paper, we develop an Asymptotic-Preserving (AP) stochastic Galerkin scheme for the radiative heat transfer equations with random inputs and diffusive scalings. In this problem the random inputs arise due to uncertainties in cross section, initial data or boundary data. We use the generalized polynomial chaos based stochastic Galerkin (gPC-SG) method, which is combined with the micro–macro decomposition based deterministic AP framework in order to handle efficiently the diffusive regime. For linearized problem we prove the regularity of the solution in the random space and consequently the spectral accuracy of the gPC-SG method. We also prove the uniform (in the mean free path) linear stability for the space-time discretizations. Several numerical tests are presented to show the efficiency and accuracy of proposed scheme, especially in the diffusive regime.},

doi = {10.1016/J.JCP.2016.12.033},

journal = {Journal of Computational Physics},

number = ,

volume = 334,

place = {United States},

year = {Sat Apr 01 00:00:00 EDT 2017},

month = {Sat Apr 01 00:00:00 EDT 2017}

}