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General relaxation methods for initial-value problems with application to multistep schemes

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Recently, an approach known as relaxation has been developed for preserving the correct evolution of a functional in the numerical solution of initial-value problems, using Runge–Kutta methods. We generalize this approach to multistep methods, including all general linear methods of order two or higher, and many other classes of schemes. We prove the existence of a valid relaxation parameter and high-order accuracy of the resulting method, in the context of general equations, including but not limited to conservative or dissipative systems. The theory is illustrated with several numerical examples.

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Acknowledgements

Research reported in this publication was supported by the King Abdullah University of Science and Technology (KAUST). The project “Application-domain-specific highly reliable IT solutions” has been implemented with the support provided from the National Research, Development and Innovation Fund of Hungary, and financed under the scheme Thematic Excellence Programme no. 2020-4.1.1-TKP2020 (National Challenges Subprogramme).

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  1. Extreme Computing Research Center (ECRC), Computer Electrical and Mathematical Science and Engineering Division (CEMSE), King Abdullah University of Science and Technology (KAUST), Thuwal, 23955-6900, Saudi Arabia

    Hendrik Ranocha & David I. Ketcheson

  2. Department of Numerical Analysis, Eötvös Loránd University, Budapest, Hungary

    Lajos Lóczi

  3. Department of Differential Equations, Budapest University of Technology and Economics, Budapest, Hungary

    Lajos Lóczi

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  1. Hendrik Ranocha
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Ranocha, H., Lóczi, L. & Ketcheson, D.I. General relaxation methods for initial-value problems with application to multistep schemes. Numer. Math. 146, 875–906 (2020). https://doi.org/10.1007/s00211-020-01158-4

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