Entropy Stable and Well-Balanced Discontinuous Galerkin Methods for the Nonlinear Shallow Water Equations

The nonlinear shallow water equations (SWEs) are widely used to model the unsteady water flows in rivers and coastal areas, with extensive applications in ocean and hydraulic engineering. In this work, we propose entropy stable, well-balanced and positivity-preserving discontinuous Galerkin (DG) met...

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Bibliographic Details
Published inJournal of scientific computing Vol. 83; no. 3; p. 66
Main Authors Wen, Xiao, Don, Wai Sun, Gao, Zhen, Xing, Yulong
Format Journal Article
LanguageEnglish
Published New York Springer US 01.06.2020
Springer Nature B.V
Subjects
Algorithms
Approximation
Coastal engineering
Coastal zone
Computational Mathematics and Numerical Analysis
Conservation laws
Entropy
Galerkin method
Hydraulic engineering
Mathematical analysis
Mathematical and Computational Engineering
Mathematical and Computational Physics
Mathematics
Mathematics and Statistics
Numerical analysis
Quadratures
Shallow water equations
Theoretical
Topography
Shallow water equations
Positivity-preserving limiter
Entropy conservative
Well-balanced property
Discontinuous Galerkin methods
Entropy stable
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Summary:The nonlinear shallow water equations (SWEs) are widely used to model the unsteady water flows in rivers and coastal areas, with extensive applications in ocean and hydraulic engineering. In this work, we propose entropy stable, well-balanced and positivity-preserving discontinuous Galerkin (DG) methods, under arbitrary choices of quadrature rules, for the SWEs with a non-flat bottom topography. In Chan (J Comput Phys 362:346–374, 2018), a SBP-like differentiation operator was introduced to construct the discretely entropy conservative DG methods. We extend this idea to the SWEs and establish an entropy stable scheme by adding additional dissipative terms. Careful approximation of the source term is included to ensure the well-balanced property of the resulting method. A simple positivity-preserving limiter, compatible with the entropy stable property, is included to guarantee the non-negative water heights during the computation. One- and two-dimensional numerical experiments are presented to demonstrate the performance of the proposed methods.
ISSN:0885-7474
1573-7691
DOI:10.1007/s10915-020-01248-3