Entropy-Conservative Discontinuous Galerkin Methods for the Shallow Water Equations with Uncertainty

doi:10.1007/s42967-024-00369-y

Communications on Applied Mathematics and Computation ›› 2024, Vol. 6 ›› Issue (3): 1978-2010.doi: 10.1007/s42967-024-00369-y

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Entropy-Conservative Discontinuous Galerkin Methods for the Shallow Water Equations with Uncertainty

Janina Bender¹, Philipp Öffner^2,3

1 Institute of Mathematics, University Kassel, Mönchebergstraße 19, 34127 Kassel, Germany;
2 Institute of Mathematics, Johannes Gutenberg University, Staudingerweg 9, 55099 Mainz, Germany;
3 Mathematical Institute, Technical University Clausthal, Erzstraße 1, 38678 Clausthal-Zellerfeld, Germany

收稿日期:2023-02-27 修回日期:2023-11-22 接受日期:2024-01-08 发布日期:2024-12-20
通讯作者: Philipp Öffner,mail@philippoeffner.de;Janina Bender,jmbender@mathematik.uni-kassel.de E-mail:mail@philippoeffner.de;jmbender@mathematik.uni-kassel.de
基金资助:
The authors like to thank Stephan Gerster for some discussion about entropy-entropy flux pairs and Haar wavelet expansion. P.Ö. also gratefully acknowledge support of the Gutenberg Research College, JGU Mainz.

Entropy-Conservative Discontinuous Galerkin Methods for the Shallow Water Equations with Uncertainty

Janina Bender¹, Philipp Öffner^2,3

1 Institute of Mathematics, University Kassel, Mönchebergstraße 19, 34127 Kassel, Germany;
2 Institute of Mathematics, Johannes Gutenberg University, Staudingerweg 9, 55099 Mainz, Germany;
3 Mathematical Institute, Technical University Clausthal, Erzstraße 1, 38678 Clausthal-Zellerfeld, Germany

Received:2023-02-27 Revised:2023-11-22 Accepted:2024-01-08 Published:2024-12-20
Contact: Philipp Öffner,mail@philippoeffner.de;Janina Bender,jmbender@mathematik.uni-kassel.de E-mail:mail@philippoeffner.de;jmbender@mathematik.uni-kassel.de
Supported by:
The authors like to thank Stephan Gerster for some discussion about entropy-entropy flux pairs and Haar wavelet expansion. P.Ö. also gratefully acknowledge support of the Gutenberg Research College, JGU Mainz.

摘要/Abstract

摘要： In this paper, we develop an entropy-conservative discontinuous Galerkin (DG) method for the shallow water (SW) equation with random inputs. One of the most popular methods for uncertainty quantification is the generalized Polynomial Chaos (gPC) approach which we consider in the following manuscript. We apply the stochastic Galerkin (SG) method to the stochastic SW equations. Using the SG approach in the stochastic hyperbolic SW system yields a purely deterministic system that is not necessarily hyperbolic anymore. The lack of the hyperbolicity leads to ill-posedness and stability issues in numerical simulations. By transforming the system using Roe variables, the hyperbolicity can be ensured and an entropy-entropy flux pair is known from a recent investigation by Gerster and Herty (Commun. Comput. Phys. 27(3): 639–671, 2020). We use this pair and determine a corresponding entropy flux potential. Then, we construct entropy conservative numerical twopoint fluxes for this augmented system. By applying these new numerical fluxes in a nodal DG spectral element method (DGSEM) with flux differencing ansatz, we obtain a provable entropy conservative (dissipative) scheme. In numerical experiments, we validate our theoretical findings.

关键词: Shallow water (SW) equations, Entropy conservation/dissipation, Uncertainty quantification, Discontinuous Galerkin (DG), Generalized Polynomial Chaos (gPC)

Abstract: In this paper, we develop an entropy-conservative discontinuous Galerkin (DG) method for the shallow water (SW) equation with random inputs. One of the most popular methods for uncertainty quantification is the generalized Polynomial Chaos (gPC) approach which we consider in the following manuscript. We apply the stochastic Galerkin (SG) method to the stochastic SW equations. Using the SG approach in the stochastic hyperbolic SW system yields a purely deterministic system that is not necessarily hyperbolic anymore. The lack of the hyperbolicity leads to ill-posedness and stability issues in numerical simulations. By transforming the system using Roe variables, the hyperbolicity can be ensured and an entropy-entropy flux pair is known from a recent investigation by Gerster and Herty (Commun. Comput. Phys. 27(3): 639–671, 2020). We use this pair and determine a corresponding entropy flux potential. Then, we construct entropy conservative numerical twopoint fluxes for this augmented system. By applying these new numerical fluxes in a nodal DG spectral element method (DGSEM) with flux differencing ansatz, we obtain a provable entropy conservative (dissipative) scheme. In numerical experiments, we validate our theoretical findings.

Key words: Shallow water (SW) equations, Entropy conservation/dissipation, Uncertainty quantification, Discontinuous Galerkin (DG), Generalized Polynomial Chaos (gPC)

中图分类号:

Janina Bender, Philipp Öffner. Entropy-Conservative Discontinuous Galerkin Methods for the Shallow Water Equations with Uncertainty[J]. Communications on Applied Mathematics and Computation, 2024, 6(3): 1978-2010.

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Entropy-Conservative Discontinuous Galerkin Methods for the Shallow Water Equations with Uncertainty

Entropy-Conservative Discontinuous Galerkin Methods for the Shallow Water Equations with Uncertainty

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