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

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 two-point 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.

Keywords

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

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Funding

Gutenberg Forschungskolleg; Johannes Gutenberg-Universit?t Mainz (1030)