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Compact finite difference schemes of arbitrary order for the Poisson equation in arbitrary dimensions

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Abstract

A formulation of the Taylor expansion with symmetric polynomial algebra allows to compute the coefficients of compact finite difference schemes, which solve the Poisson equation at an arbitrary order of accuracy on a uniform Cartesian grid in arbitrary dimensions. This construction produces original high order schemes which respect the Discrete Maximum Principle: a tenth order scheme in dimension three and several sixth order schemes in arbitrary dimension. Numerical experiments validate the accuracy of these schemes.

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Acknowledgements

The author would like to thank the editor and referees for their valuable comments and suggestions which helped to improve the clarity of this manuscript and to enhance its results. He is also grateful to Jean-Pierre Croisille for informative discussions and for his help in provi ding a rich bibliography.

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  1. Institut Jean Lamour, Matériaux-Métallurgie-Nanosciences-Plasmas-Surfaces, UMR 7198 - CNRS, Faculté des Sciences et Technologies, Université de Lorraine, Campus Victor Grignard, BP 70239, 54506, Vandoeuvre-les-Nancy Cedex, France

    Erwan Deriaz

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Correspondence to Erwan Deriaz.

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Deriaz, E. Compact finite difference schemes of arbitrary order for the Poisson equation in arbitrary dimensions. Bit Numer Math 60, 199–233 (2020). https://doi.org/10.1007/s10543-019-00772-5

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  • DOI: https://doi.org/10.1007/s10543-019-00772-5

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