# A stochastic Galerkin method for first-order quasilinear hyperbolic systems with uncertainty

## Abstract

This paper is concerned with generalized polynomial chaos (gPC) approximation for first-order quasilinear hyperbolic systems with uncertainty. The one-dimensional (1D) hyperbolic system is first symmetrized with the aid of left eigenvector matrix of the Jacobian matrix. Then the gPC stochastic Galerkin method is applied to derive a provably symmetrically hyperbolic equations for the gPC expansion coefficients. The resulting deterministic gPC Galerkin system is discretized by a path-conservative finite volume WENO scheme in space and a third-order total variation diminishing Runge–Kutta method in time. The method is further extended to two-dimensional (2D) quasilinear hyperbolic system with uncertainty, where the symmetric hyperbolicity of the one-dimensional gPC Galerkin system is carried over via an operator splitting technique. Several numerical experiments are conducted to demonstrate the accuracy and effectiveness of the proposed gPC stochastic Galerkin method.

- Authors:

- HEDPS, CAPT & LMAM, School of Mathematical Sciences, Peking University, Beijing 100871 (China)
- (China)
- Department of Mathematics, The Ohio State University, Columbus, OH 43210 (United States)

- Publication Date:

- OSTI Identifier:
- 22701596

- Resource Type:
- Journal Article

- Journal Name:
- Journal of Computational Physics

- Additional Journal Information:
- Journal Volume: 345; Other Information: Copyright (c) 2017 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; CHAOS THEORY; MATHEMATICAL OPERATORS; MATRICES; ONE-DIMENSIONAL CALCULATIONS; RUNGE-KUTTA METHOD; STOCHASTIC PROCESSES; TWO-DIMENSIONAL CALCULATIONS

### Citation Formats

```
Wu, Kailiang, Tang, Huazhong, School of Mathematics and Computational Science, Xiangtan University, Xiangtan 411105, Hunan Province, and Xiu, Dongbin.
```*A stochastic Galerkin method for first-order quasilinear hyperbolic systems with uncertainty*. United States: N. p., 2017.
Web. doi:10.1016/J.JCP.2017.05.027.

```
Wu, Kailiang, Tang, Huazhong, School of Mathematics and Computational Science, Xiangtan University, Xiangtan 411105, Hunan Province, & Xiu, Dongbin.
```*A stochastic Galerkin method for first-order quasilinear hyperbolic systems with uncertainty*. United States. doi:10.1016/J.JCP.2017.05.027.

```
Wu, Kailiang, Tang, Huazhong, School of Mathematics and Computational Science, Xiangtan University, Xiangtan 411105, Hunan Province, and Xiu, Dongbin. Fri .
"A stochastic Galerkin method for first-order quasilinear hyperbolic systems with uncertainty". United States. doi:10.1016/J.JCP.2017.05.027.
```

```
@article{osti_22701596,
```

title = {A stochastic Galerkin method for first-order quasilinear hyperbolic systems with uncertainty},

author = {Wu, Kailiang and Tang, Huazhong and School of Mathematics and Computational Science, Xiangtan University, Xiangtan 411105, Hunan Province and Xiu, Dongbin},

abstractNote = {This paper is concerned with generalized polynomial chaos (gPC) approximation for first-order quasilinear hyperbolic systems with uncertainty. The one-dimensional (1D) hyperbolic system is first symmetrized with the aid of left eigenvector matrix of the Jacobian matrix. Then the gPC stochastic Galerkin method is applied to derive a provably symmetrically hyperbolic equations for the gPC expansion coefficients. The resulting deterministic gPC Galerkin system is discretized by a path-conservative finite volume WENO scheme in space and a third-order total variation diminishing Runge–Kutta method in time. The method is further extended to two-dimensional (2D) quasilinear hyperbolic system with uncertainty, where the symmetric hyperbolicity of the one-dimensional gPC Galerkin system is carried over via an operator splitting technique. Several numerical experiments are conducted to demonstrate the accuracy and effectiveness of the proposed gPC stochastic Galerkin method.},

doi = {10.1016/J.JCP.2017.05.027},

journal = {Journal of Computational Physics},

issn = {0021-9991},

number = ,

volume = 345,

place = {United States},

year = {2017},

month = {9}

}