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Numerische Mathematik

, Volume 138, Issue 4, pp 939–973 | Cite as

Well-balanced schemes for the shallow water equations with Coriolis forces

  • Alina Chertock
  • Michael Dudzinski
  • Alexander Kurganov
  • Mária Lukáčová-Medvid’ová
Article

Abstract

In the present paper we study shallow water equations with bottom topography and Coriolis forces. The latter yield non-local potential operators that need to be taken into account in order to derive a well-balanced numerical scheme. In order to construct a higher order approximation a crucial step is a well-balanced reconstruction which has to be combined with a well-balanced update in time. We implement our newly developed second-order reconstruction in the context of well-balanced central-upwind and finite-volume evolution Galerkin schemes. Theoretical proofs and numerical experiments clearly demonstrate that the resulting finite-volume methods preserve exactly the so-called jets in the rotational frame. For general two-dimensional geostrophic equilibria the well-balanced methods, while not preserving the equilibria exactly, yield better resolution than their non-well-balanced counterparts.

Mathematics Subject Classification

65L05 65M06 35L45 35L65 65M25 65M15 

Notes

Acknowledgements

The work of A. Chertock was supported in part by the NSF Grants DMS-1216974 and DMS-1521051 and the ONR Grant N00014-12-1-0832. The work of A. Kurganov was supported in part by the NSF Grants DMS-1216957 and DMS-1521009 and the ONR Grant N00014-12-1-0833. M. Lukáčová was supported by the German Science Foundation (DFG) Grants LU 1470/2-3 and SFB TRR 165 “Waves to Weather”. We would like to thank Doron Levy (University of Maryland) and Leonid Yelash (University of Mainz) for fruitful discussions on the topic.

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Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2017

Authors and Affiliations

  • Alina Chertock
    • 1
  • Michael Dudzinski
    • 2
  • Alexander Kurganov
    • 3
    • 4
  • Mária Lukáčová-Medvid’ová
    • 5
  1. 1.Department of MathematicsNorth Carolina State UniversityRaleighUSA
  2. 2.Department of the Theory of Electrical EngineeringHelmut Schmidt University of the Federal Armed Forces HamburgHamburgGermany
  3. 3.Department of MathematicsSouthern University of Science and Technology of ChinaShenzhenChina
  4. 4.Mathematics DepartmentTulane UniversityNew OrleansUSA
  5. 5.Institute of MathematicsUniversity of MainzMainzGermany

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