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

TL;DR

This paper introduces a novel two-parameter family of entropy conservative fluxes for shallow water equations, ensuring well-balanced, entropy stable, and positivity-preserving numerical methods using SBP operators and finite volume subcells.

Contribution

It develops a new two-parameter family of entropy conservative fluxes for shallow water equations that are well-balanced, entropy stable, and positivity preserving, with comprehensive analysis and numerical tests.

Findings

New entropy conservative fluxes preserve lake-at-rest state.

Enhanced schemes ensure positivity and stability.

Numerical tests confirm effectiveness of proposed methods.

Abstract

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 Gau{\ss} nodes. 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). 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.