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Journal of Scientific Computing
Published in: Journal of Scientific Computing 3/2020

01-06-2020

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

Authors: Xiao Wen, Wai Sun Don, Zhen Gao, Yulong Xing

Published in: Journal of Scientific Computing | Issue 3/2020

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Abstract

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.
previous article Linearly Implicit and High-Order Energy-Conserving Schemes for Nonlinear Wave Equations
next article A High Accuracy Nonconforming Finite Element Scheme for Helmholtz Transmission Eigenvalue Problem

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Appendix
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Metadata
Title
Entropy Stable and Well-Balanced Discontinuous Galerkin Methods for the Nonlinear Shallow Water Equations
Authors
Xiao Wen
Wai Sun Don
Zhen Gao
Yulong Xing
Publication date
01-06-2020
Publisher
Springer US
Published in
Journal of Scientific Computing / Issue 3/2020
Print ISSN: 0885-7474
Electronic ISSN: 1573-7691
DOI
https://doi.org/10.1007/s10915-020-01248-3

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