Entropy Stable Galerkin Methods with Suitable Quadrature Rules for Hyperbolic Systems with Random Inputs
Entropy Stable Galerkin Methods with Suitable Quadrature Rules for Hyperbolic Systems with Random Inputs
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DOI：
10.1007/s10915-022-01866-z
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发表时间：
2022-06
影响因子：
2.5
通讯作者：
Xinghui Zhong;Chi-Wang Shu
中科院分区：
文献类型：
--
作者：
Xinghui Zhong;Chi-Wang Shu
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.
