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Mimetic properties of difference operators: product and chain rules as for functions of bounded variation and entropy stability of second derivatives

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For discretisations of hyperbolic conservation laws, mimicking properties of operators or solutions at the continuous (differential equation) level discretely has resulted in several successful methods. While well-posedness for nonlinear systems in several space dimensions is an open problem, mimetic properties such as summation-by-parts as discrete analogue of integration-by-parts allow a direct transfer of some results and their proofs, e.g. stability for linear systems. In this article, discrete analogues of the generalised product and chain rules that apply to functions of bounded variation are considered. It is shown that such analogues hold for certain second order operators but are not possible for higher order approximations. Furthermore, entropy dissipation by second derivatives with varying coefficients is investigated, showing again the far stronger mimetic properties of second order approximations compared to higher order ones.

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Acknowledgements

This work was supported by the German Research Foundation (DFG, Deutsche Forschungsgemeinschaft) under Grant SO 363/14-1. The author would like to thank the anonymous reviewers for their helpful comments and valuable suggestions to improve this article.

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    Hendrik Ranocha

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Communicated by Jan Nordström.

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Ranocha, H. Mimetic properties of difference operators: product and chain rules as for functions of bounded variation and entropy stability of second derivatives. Bit Numer Math 59, 547–563 (2019). https://doi.org/10.1007/s10543-018-0736-7

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