Abstract
An efficient approximate version of implicit Taylor methods for initial-value problems of systems of ordinary differential equations (ODEs) is introduced. The approach, based on an approximate formulation of Taylor methods, produces a method that requires less evaluations of the function that defines the ODE and its derivatives than the usual version. On the other hand, an efficient numerical solution of the equation that arises from the discretization by means of Newton’s method is introduced for an implicit scheme of any order. Numerical experiments illustrate that the resulting algorithm is simpler to implement and has better performance than its exact counterpart.
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Acknowledgements
A.B., M.C.M., and P.M. are supported by Spanish MINECO grant MTM2017-83942-P. P.M. is also supported by Conicyt/ANID (Chile), project PAI-MEC, folio 80150006. R.B. is supported by Fondecyt project 1170473; CRHIAM, Proyecto ANID/Fondap/15130015; Basal project CONICYT/PIA/AFB170001; and by the INRIA Associated Team “Efficient numerical schemes for non-local transport phenomena” (NOLOCO; 2018–2020), and D.Z. is supported by Conicyt/ANID Fondecyt/Postdoctorado/3170077.
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Baeza, A., Bürger, R., Martí, M.d.C. et al. On approximate implicit Taylor methods for ordinary differential equations. Comp. Appl. Math. 39, 304 (2020). https://doi.org/10.1007/s40314-020-01356-8
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DOI: https://doi.org/10.1007/s40314-020-01356-8