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

作　　者：Janina Bender Philipp Öffner 

机构地区：[1]Institute of Mathematics,University Kassel,Monchebergstraße 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

出　　处：《Communications on Applied Mathematics and Computation》2024年第3期1978-2010,共33页应用数学与计算数学学报（英文）

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 quantifcation 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 fux 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 fux potential.Then,we construct entropy conservative numerical twopoint fuxes for this augmented system.By applying these new numerical fuxes in a nodal DG spectral element method(DGSEM)with fux diferencing ansatz,we obtain a provable entropy conservative(dissipative)scheme.In numerical experiments,we validate our theoretical fndings.

关 键 词：Shallow water(SW)equations Entropy conservation/dissipation Uncertainty quantification Discontinuous Galerkin(DG) Generalized Polynomial Chaos(gPC) 

分 类 号：O175[理学—数学]