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Entropy Stable Galerkin Methods with Suitable Quadrature Rules for Hyperbolic Systems with Random Inputs

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Abstract

In this paper, we investigate hyperbolic systems with random inputs based on generalized polynomial chaos (gPC) approximations, which is one of the most popular methods for uncertainty quantification (UQ) and can be implemented with either the stochastic Galerkin (SG) method or the stochastic collocation (SC) method. One of the challenges for solving stochastic hyperbolic systems with the SG method is that the resulting deterministic system may not be hyperbolic. The lack of hyperbolicity may lead to the ill-posedness of the problem and the instability of numerical simulations. The main objective of this paper is to show that by approximating the solution in the random space with the SG method in a pseudo-spectral way with suitable quadrature rules, the SG scheme can be written as a SC scheme on a set of specific nodes. The resulting collocation scheme preserves the hyperbolicity of the original hyperbolic system, and is more efficient to implement. On the other hand, entropy conditions play an essential role in the well-posedness of hyperbolic conservation laws. Thus we approximate the resulted collocation scheme in space by the entropy stable nodal discontinuous Galerkin (DG) method Chen and Shu (J. Comput. Phys. 345:427-461, 2017), where the entropy stability is guaranteed by high order summation-by-parts operators, entropy conservative fluxes and entropy stable fluxes. Numerical experiments are performed to validate the accuracy and effectiveness of the proposed numerical red schemes.

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Acknowledgements

X. Zhong was partially supported by the National Natural Science Foundation of China (NSFC) (Grant No. 11871428). C.-W. Shu was partially supported by NSF grant DMS-2010107 and AFOSR grant FA9550-20-1-0055. The authors would like to thank Dr. Tianheng Chen for many helpful discussions in the computation.

Funding

Author X. Zhong was partially supported by the National Natural Science Foundation of China (NSFC) (Grant No. 11871428). Author C.-W. Shu was partially supported by NSF grant DMS-2010107 and AFOSR grant FA9550-20-1-0055.

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Authors and Affiliations

  1. School of Mathematical Sciences, Zhejiang University, 310027, Hangzhou, China

    Xinghui Zhong

  2. Division of Applied Mathematics, Brown University, 02912, Providence, RI, USA

    Chi-Wang Shu

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  1. Xinghui Zhong
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Correspondence to Xinghui Zhong.

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Zhong, X., Shu, CW. Entropy Stable Galerkin Methods with Suitable Quadrature Rules for Hyperbolic Systems with Random Inputs. J Sci Comput 92, 14 (2022). https://doi.org/10.1007/s10915-022-01866-z

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