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

Hendrik Ranocha

Published 2016-09-26Version 1

Entropy stable semidiscretisations of the shallow water equations are developed, based on summation-by-parts (SBP) operators and using split forms of the equations. The resulting two-parameter family of entropy conservative schemes for general SBP bases, especially using Gau{\ss} nodes, is adapted to varying bottom topography in a well-balanced way, i.e. preserving the lake-at-rest steady state. Moreover, positivity preservation is ensured using the framework of Zhang and Shu (Maximum-principle-satisfying and positivity-preserving high-order schemes for conservation laws: survey and recent developments, 2011. In: Proceedings of the Royal Society of London A: Mathematical, Physical and Engineering Sciences, The Royal Society, vol 467, pp. 2752--2766) and finite volume subcells, adapted to nodal SBP bases with diagonal mass matrix. Numerical tests of the proposed schemes are performed and some conclusions are presented.