Shallow water equations: split-form, entropy stable, well-balanced, and positivity preserving numerical methods

  • Hendrik RanochaEmail author
Original Paper


For the first time, a general two-parameter family of entropy conservative numerical fluxes for the shallow water equations is developed and investigated. These are adapted to a varying bottom topography in a well-balanced way, i.e. preserving the lake-at-rest steady state. Furthermore, these fluxes are used to create entropy stable and well-balanced split-form semidiscretisations based on general summation-by-parts (SBP) operators, including Gauß nodes. Moreover, positivity preservation is ensured using the framework of Zhang and Shu (Proc R Soc Lond A Math Phys Eng Sci 467: 2752–2776, 2011). Therefore, the new two-parameter family of entropy conservative fluxes is enhanced by dissipation operators and investigated with respect to positivity preservation. Additionally, some known entropy stable and positive numerical fluxes are compared. Furthermore, finite volume subcells adapted to nodal SBP bases with diagonal mass matrix are used. Finally, numerical tests of the proposed schemes are performed and some conclusions are presented.


Skew-symmetric shallow water equations Summation-by-parts Split-form Entropy stability Well-balancedness Positivity preservation 

Mathematics Subject Classification

65M70 65M60 65M06 65M12 



The author would like to thank the anonymous reviewer for some helpful comments, resulting in an improved presentation of this material.


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Copyright information

© Springer-Verlag Berlin Heidelberg 2016

Authors and Affiliations

  1. 1.TU BraunschweigBrunswickGermany

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