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A well-designed numerical method for the shallow water equations (SWE) shouldensure well-balancedness, nonnegativity of water heights, and entropystability. For a continuous finite element discretization of a nonlinearhyperbolic system without source terms, positivity preservation and entropystability can be enforced using the framework of algebraic flux correction(AFC). In this work, we develop a well-balanced AFC scheme for the SWE systemincluding a topography source term. Our method preserves the lake at restequilibrium up to machine precision. The low-order version represents ageneralization of existing finite volume approaches to the finite elementsetting. The high-order extension is equipped with a property-preserving fluxlimiter. Nonnegativity of water heights is guaranteed under a standard CFLcondition. Moreover, the flux-corrected space discretization satisfies asemi-discrete entropy inequality. New algorithms are proposed for realisticsimulation of wetting and drying processes. Numerical examples for well-knownbenchmarks are presented to evaluate the performance of the scheme.

Bound-preserving and entropy-stable algebraic flux correction schemes for the shallow water equations with topography