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For the first time, a general two-parameter family of entropy conservativenumerical 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 tocreate entropy stable and well-balanced split-form semidiscretisations based ongeneral summation-by-parts (SBP) operators, including Gau{\ss} nodes. Moreover,positivity preservation is ensured using the framework of Zhang and Shu(Maximum-principle-satisfying and positivity-preserving high-order schemes forconservation laws: survey and recent developments, 2011. In: Proceedings of theRoyal Society of London A: Mathematical, Physical and Engineering Sciences, TheRoyal Society, vol 467, pp. 2752--2766). Therefore, the new two-parameterfamily of entropy conservative fluxes is enhanced by dissipation operators andinvestigated with respect to positivity preservation. Additionally, some knownentropy stable and positive numerical fluxes are compared. Furthermore, finitevolume subcells adapted to nodal SBP bases with diagonal mass matrix are used.Finally, numerical tests of the proposed schemes are performed and someconclusions are presented.

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