Abstract
Passive and active Richardson extrapolations are robust devices to increase the rate of convergence of time integration methods. While the order of convergence is shown to increase by one under rather natural smoothness conditions if the passive Richardson extrapolation is used, for the active Richardson extrapolation the increase of the order has not been generally proven. It is known that the Lipschitz property of the right-hand side function of the differential equation to be solved yields convergence of order p if the method is consistent in order p. In this paper it is shown that the active Richardson extrapolation increases the order of consistency by one when the underlying method is any Runge–Kutta method of order \(p=1,2\), or 3.
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Acknowledgements
“Application Domain Specific Highly Reliable IT Solutions” project has been implemented with the support provided from the National Research, Development and Innovation Fund of Hungary, financed under the Thematic Excellence Programme TKP2020-NKA-06 (National Challenges Subprogramme) funding scheme. This work was completed in the ELTE Institutional Excellence Program (TKP2020-IKA-05) financed by the Hungarian Ministry of Human Capacities. The project has been supported by the European Union, and co-financed by the European Social Fund (EFOP-3.6.3-VEKOP-16-2017-00002), and further, it was supported by the Hungarian Scientific Research Fund OTKA SNN125119.
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Bayleyegn, T., Faragó, I., Havasi, Á. (2022). On the Consistency Order of Runge–Kutta Methods Combined with Active Richardson Extrapolation. In: Lirkov, I., Margenov, S. (eds) Large-Scale Scientific Computing. LSSC 2021. Lecture Notes in Computer Science, vol 13127. Springer, Cham. https://doi.org/10.1007/978-3-030-97549-4_11
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DOI: https://doi.org/10.1007/978-3-030-97549-4_11
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