# Asymptotic-preserving methods for hyperbolic and transport equations with random inputs and diffusive scalings

## Abstract

In this paper we develop a set of stochastic numerical schemes for hyperbolic and transport equations with diffusive scalings and subject to random inputs. The schemes are asymptotic preserving (AP), in the sense that they preserve the diffusive limits of the equations in discrete setting, without requiring excessive refinement of the discretization. Our stochastic AP schemes are extensions of the well-developed deterministic AP schemes. To handle the random inputs, we employ generalized polynomial chaos (gPC) expansion and combine it with stochastic Galerkin procedure. We apply the gPC Galerkin scheme to a set of representative hyperbolic and transport equations and establish the AP property in the stochastic setting. We then provide several numerical examples to illustrate the accuracy and effectiveness of the stochastic AP schemes.

- Authors:

- Department of Mathematics, Institute of Natural Sciences and MOE-LSC, Shanghai Jiao Tong University, Shanghai 200240 (China)
- (United States)
- Scientific Computing and Imaging Institute, University of Utah, Salt Lake City, UT 84112 (United States)

- Publication Date:

- OSTI Identifier:
- 22465624

- Resource Type:
- Journal Article

- Journal Name:
- Journal of Computational Physics

- Additional Journal Information:
- Journal Volume: 289; Other Information: Copyright (c) 2015 Elsevier Science B.V., Amsterdam, The Netherlands, All rights reserved.; Country of input: International Atomic Energy Agency (IAEA); Journal ID: ISSN 0021-9991

- Country of Publication:
- United States

- Language:
- English

- Subject:
- 71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSICS; ACCURACY; ASYMPTOTIC SOLUTIONS; CHAOS THEORY; POLYNOMIALS; RANDOMNESS; STOCHASTIC PROCESSES; TRANSPORT THEORY

### Citation Formats

```
Jin, Shi, E-mail: jin@math.wisc.edu, Department of Mathematics, University of Wisconsin, Madison, WI 53706, Xiu, Dongbin, E-mail: dongbin.xiu@utah.edu, Department of Mathematics, University of Utah, Salt Lake City, UT 84112, and Zhu, Xueyu, E-mail: xzhu@sci.utah.edu.
```*Asymptotic-preserving methods for hyperbolic and transport equations with random inputs and diffusive scalings*. United States: N. p., 2015.
Web. doi:10.1016/J.JCP.2015.02.023.

```
Jin, Shi, E-mail: jin@math.wisc.edu, Department of Mathematics, University of Wisconsin, Madison, WI 53706, Xiu, Dongbin, E-mail: dongbin.xiu@utah.edu, Department of Mathematics, University of Utah, Salt Lake City, UT 84112, & Zhu, Xueyu, E-mail: xzhu@sci.utah.edu.
```*Asymptotic-preserving methods for hyperbolic and transport equations with random inputs and diffusive scalings*. United States. doi:10.1016/J.JCP.2015.02.023.

```
Jin, Shi, E-mail: jin@math.wisc.edu, Department of Mathematics, University of Wisconsin, Madison, WI 53706, Xiu, Dongbin, E-mail: dongbin.xiu@utah.edu, Department of Mathematics, University of Utah, Salt Lake City, UT 84112, and Zhu, Xueyu, E-mail: xzhu@sci.utah.edu. Fri .
"Asymptotic-preserving methods for hyperbolic and transport equations with random inputs and diffusive scalings". United States. doi:10.1016/J.JCP.2015.02.023.
```

```
@article{osti_22465624,
```

title = {Asymptotic-preserving methods for hyperbolic and transport equations with random inputs and diffusive scalings},

author = {Jin, Shi, E-mail: jin@math.wisc.edu and Department of Mathematics, University of Wisconsin, Madison, WI 53706 and Xiu, Dongbin, E-mail: dongbin.xiu@utah.edu and Department of Mathematics, University of Utah, Salt Lake City, UT 84112 and Zhu, Xueyu, E-mail: xzhu@sci.utah.edu},

abstractNote = {In this paper we develop a set of stochastic numerical schemes for hyperbolic and transport equations with diffusive scalings and subject to random inputs. The schemes are asymptotic preserving (AP), in the sense that they preserve the diffusive limits of the equations in discrete setting, without requiring excessive refinement of the discretization. Our stochastic AP schemes are extensions of the well-developed deterministic AP schemes. To handle the random inputs, we employ generalized polynomial chaos (gPC) expansion and combine it with stochastic Galerkin procedure. We apply the gPC Galerkin scheme to a set of representative hyperbolic and transport equations and establish the AP property in the stochastic setting. We then provide several numerical examples to illustrate the accuracy and effectiveness of the stochastic AP schemes.},

doi = {10.1016/J.JCP.2015.02.023},

journal = {Journal of Computational Physics},

issn = {0021-9991},

number = ,

volume = 289,

place = {United States},

year = {2015},

month = {5}

}