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Entropy Stable Flux Correction for Scalar Hyperbolic Conservation Laws

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Abstract

It is known that Flux Corrected Transport algorithms can produce entropy-violating solutions of hyperbolic conservation laws. Our purpose is to design flux correction with maximal antidiffusive fluxes to obtain entropy solutions of scalar hyperbolic conservation laws. To do this we consider a hybrid difference scheme that is a linear combination of a monotone scheme and a scheme of high-order accuracy. 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 monotone 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 of scalar hyperbolic conservation laws. A nontrivial approximate solution of the optimization problem yields expressions to compute the required flux limiters. We present examples that show that not all numerical entropy fluxes guarantee to single out a physically correct solution of scalar hyperbolic conservation laws.

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Data Availability Statement

Enquiries about data availability should be directed to the authors.

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Acknowledgements

We thank anonymous reviewers for their detailed comments, which helped us to significantly improve the paper.

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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 Scalar Hyperbolic Conservation Laws. J Sci Comput 91, 10 (2022). https://doi.org/10.1007/s10915-022-01792-0

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