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Arbitrary high order A-stable and B-convergent numerical methods for ODEs via deferred correction

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Abstract

This paper presents a sequence of deferred correction (DC) schemes built recursively from the implicit midpoint scheme for the numerical solution of general first order ordinary differential equations (ODEs). It is proven that each scheme is A-stable, satisfies a B-convergence property, and that the correction on a scheme DC2j of order 2j of accuracy leads to a scheme DC2j + 2 of order 2j + 2. The order of accuracy is guaranteed by a deferred correction condition. Numerical experiments with standard stiff and non-stiff ODEs are performed with the DC2, ..., DC10 schemes. The results show a high accuracy of the method. The theoretical orders of accuracy are achieved together with a satisfactory stability.

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Authors and Affiliations

  1. Department of Mathematics and Statistics, University of Ottawa, STEM Complex, 150 Louis-Pasteur Pvt, Ottawa, ON, K1N 6N5, Canada

    Saint-Cyr E. R. Koyaguerebo-Imé & Yves Bourgault

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  1. Saint-Cyr E. R. Koyaguerebo-Imé
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  2. Yves Bourgault
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Correspondence to Yves Bourgault.

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Communicated by Antonella Zanna Munthe-Kaas.

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The authors would like to acknowledge the financial support of the Discovery Grant Program of the Natural Sciences and Engineering Research Council of Canada (NSERC) and a scholarship to the first author from the NSERC CREATE program “Génie par la Simulation”.

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Koyaguerebo-Imé, SC.E.R., Bourgault, Y. Arbitrary high order A-stable and B-convergent numerical methods for ODEs via deferred correction. Bit Numer Math 62, 139–170 (2022). https://doi.org/10.1007/s10543-021-00875-y

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