# A stochastic Galerkin method for the Boltzmann equation with uncertainty

## Abstract

We develop a stochastic Galerkin method for the Boltzmann equation with uncertainty. The method is based on the generalized polynomial chaos (gPC) approximation in the stochastic Galerkin framework, and can handle random inputs from collision kernel, initial data or boundary data. We show that a simple singular value decomposition of gPC related coefficients combined with the fast Fourier-spectral method (in velocity space) allows one to compute the high-dimensional collision operator very efficiently. In the spatially homogeneous case, we first prove that the analytical solution preserves the regularity of the initial data in the random space, and then use it to establish the spectral accuracy of the proposed stochastic Galerkin method. Several numerical examples are presented to illustrate the validity of the proposed scheme.

- Authors:

- Department of Mathematics, Purdue University, West Lafayette, IN 47907 (United States)
- Department of Mathematics, University of Wisconsin–Madison, Madison, WI 53706 (United States)
- (China)

- Publication Date:

- OSTI Identifier:
- 22572317

- Resource Type:
- Journal Article

- Resource Relation:
- Journal Name: Journal of Computational Physics; Journal Volume: 315; 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:
- 71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSICS; ACCURACY; ANALYTICAL SOLUTION; APPROXIMATIONS; BOLTZMANN EQUATION; CHAOS THEORY; COLLISIONS; KERNELS; POLYNOMIALS; RANDOMNESS; STOCHASTIC PROCESSES; VELOCITY

### Citation Formats

```
Hu, Jingwei, E-mail: jingweihu@purdue.edu, Jin, Shi, E-mail: sjin@wisc.edu, and Department of Mathematics, Institute of Natural Sciences, MOE-LSEC and SHL-MAC, Shanghai Jiao Tong University, Shanghai 200240.
```*A stochastic Galerkin method for the Boltzmann equation with uncertainty*. United States: N. p., 2016.
Web. doi:10.1016/J.JCP.2016.03.047.

```
Hu, Jingwei, E-mail: jingweihu@purdue.edu, Jin, Shi, E-mail: sjin@wisc.edu, & Department of Mathematics, Institute of Natural Sciences, MOE-LSEC and SHL-MAC, Shanghai Jiao Tong University, Shanghai 200240.
```*A stochastic Galerkin method for the Boltzmann equation with uncertainty*. United States. doi:10.1016/J.JCP.2016.03.047.

```
Hu, Jingwei, E-mail: jingweihu@purdue.edu, Jin, Shi, E-mail: sjin@wisc.edu, and Department of Mathematics, Institute of Natural Sciences, MOE-LSEC and SHL-MAC, Shanghai Jiao Tong University, Shanghai 200240. Wed .
"A stochastic Galerkin method for the Boltzmann equation with uncertainty". United States.
doi:10.1016/J.JCP.2016.03.047.
```

```
@article{osti_22572317,
```

title = {A stochastic Galerkin method for the Boltzmann equation with uncertainty},

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

abstractNote = {We develop a stochastic Galerkin method for the Boltzmann equation with uncertainty. The method is based on the generalized polynomial chaos (gPC) approximation in the stochastic Galerkin framework, and can handle random inputs from collision kernel, initial data or boundary data. We show that a simple singular value decomposition of gPC related coefficients combined with the fast Fourier-spectral method (in velocity space) allows one to compute the high-dimensional collision operator very efficiently. In the spatially homogeneous case, we first prove that the analytical solution preserves the regularity of the initial data in the random space, and then use it to establish the spectral accuracy of the proposed stochastic Galerkin method. Several numerical examples are presented to illustrate the validity of the proposed scheme.},

doi = {10.1016/J.JCP.2016.03.047},

journal = {Journal of Computational Physics},

number = ,

volume = 315,

place = {United States},

year = {Wed Jun 15 00:00:00 EDT 2016},

month = {Wed Jun 15 00:00:00 EDT 2016}

}