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Entropy Stable Flux Correction for Hydrostatic Reconstruction Scheme for Shallow Water Flows

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Abstract

First-order hydrostatic reconstruction (HR) schemes for shallow water equations are highly diffusive whereas high-order schemes can produce entropy-violating solutions. Our goal is to develop a flux correction with maximum antidiffusive fluxes to obtain entropy solutions of shallow water equations with variable bottom topography. For this purpose, we consider a hybrid explicit HR scheme whose flux is a convex combination of first-order Rusanov flux and high-order flux. The conditions under which the explicit first-order HR scheme for shallow water equations satisfies the fully discrete entropy inequality have been studied. The flux limiters for the hybrid scheme are calculated from a corresponding optimization problem. Constraints for the optimization problem consist of inequalities that are valid for the first-order HR scheme and applied to the hybrid scheme. We apply the discrete cell entropy inequality with the proper numerical entropy flux to single out a physically relevant solution to the shallow water equations. A nontrivial approximate solution of the optimization problem yields expressions to compute the required flux limiters. Numerical results of testing various HR schemes on different benchmarks are presented.

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Acknowledgements

The author is grateful to anonymous reviewers for their comments and suggestions that helped to improve the paper.

Funding

This work was supported by the National Academy of Sciences of Ukraine, project No. 0119U001433.

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  1. Institute of Mathematical Machines and System Problems, Ukrainian National Academy of Sciences, Kyiv, Ukraine

    Sergii Kivva

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Kivva, S. Entropy Stable Flux Correction for Hydrostatic Reconstruction Scheme for Shallow Water Flows. J Sci Comput 99, 1 (2024). https://doi.org/10.1007/s10915-024-02457-w

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