Universität zu Köln

Wintermeyer, Niklas, Winters, Andrew R., Gassner, Gregor J. and Kopriva, David A. (2017). An entropy stable nodal discontinuous Galerkin method for the two dimensional shallow water equations on unstructured curvilinear meshes with discontinuous bathymetry. J. Comput. Phys., 340. S. 200 - 243. SAN DIEGO: ACADEMIC PRESS INC ELSEVIER SCIENCE. ISSN 1090-2716

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Abstract

We design an arbitrary high-order accurate nodal discontinuous Galerkin spectral element approximation for the non-linear two dimensional shallow water equations with non-constant, possibly discontinuous, bathymetry on unstructured, possibly curved, quadrilateral meshes. The scheme is derived from an equivalent flux differencing formulation of the split form of the equations. We prove that this discretization exactly preserves the local mass and momentum. Furthermore, combined with a special numerical interface flux function, the method exactly preserves the mathematical entropy, which is the total energy for the shallow water equations. By adding a specific form of interface dissipation to the baseline entropy conserving scheme we create a provably entropy stable scheme. That is, the numerical scheme discretely satisfies the second law of thermodynamics. Finally, with a particular discretization of the bathymetry source term we prove that the numerical approximation is well-balanced. We provide numerical examples that verify the theoretical findings and furthermore provide an application of the scheme for a partial break of a curved dam test problem. (C) 2017 Elsevier Inc. All rights reserved.

Item Type:	Journal Article
Creators:	Creators Email ORCID ORCID Put Code Wintermeyer, Niklas UNSPECIFIED UNSPECIFIED UNSPECIFIED Winters, Andrew R. UNSPECIFIED UNSPECIFIED UNSPECIFIED Gassner, Gregor J. UNSPECIFIED UNSPECIFIED UNSPECIFIED Kopriva, David A. UNSPECIFIED UNSPECIFIED UNSPECIFIED
URN:	urn:nbn:de:hbz:38-227092
DOI:	10.1016/j.jcp.2017.03.036
Journal or Publication Title:	J. Comput. Phys.
Volume:	340
Page Range:	S. 200 - 243
Date:	2017
Publisher:	ACADEMIC PRESS INC ELSEVIER SCIENCE
Place of Publication:	SAN DIEGO
ISSN:	1090-2716
Language:	English
Faculty:	Unspecified
Divisions:	Unspecified
Subjects:	no entry
Uncontrolled Keywords:	Keywords Language FINITE-DIFFERENCE FORMULATIONS; NONLINEAR CONSERVATION-LAWS; NAVIER-STOKES EQUATIONS; SPECTRAL ELEMENT METHOD; SCHEMES; FLOWS; TERMS; DISCRETIZATION; FLUX Multiple languages Computer Science, Interdisciplinary Applications; Physics, Mathematical Multiple languages
Refereed:	Yes
URI:	http://kups.ub.uni-koeln.de/id/eprint/22709