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Geometrically adaptive grids for stiff Cauchy problems

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Abstract

A new method for automatic step size selection in the numerical integration of the Cauchy problem for ordinary differential equations is proposed. The method makes use of geometric characteristics (curvature and slope) of an integral curve. For grids generated by this method, a mesh refinement procedure is developed that makes it possible to apply the Richardson method and to obtain a posteriori asymptotically precise estimate for the error of the resulting solution (no such estimates are available for traditional step size selection algorithms). Accordingly, the proposed methods are more robust and accurate than previously known algorithms. They are especially efficient when applied to highly stiff problems, which is illustrated by numerical examples.

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Authors and Affiliations

  1. Keldysh Institute of Applied Mathematics, Russian Academy of Sciences, Miusskaya pl. 4, Moscow, 125047, Russia

    A. A. Belov, N. N. Kalitkin & I. P. Poshivaylo

  2. Faculty of Physics, Moscow State University, Moscow, 119992, Russia

    A. A. Belov

Authors
  1. A. A. Belov
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  2. N. N. Kalitkin
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  3. I. P. Poshivaylo
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Corresponding author

Correspondence to N. N. Kalitkin.

Additional information

Original Russian Text © A.A. Belov, N.N. Kalitkin, I.P. Poshivaylo, 2016, published in Doklady Akademii Nauk, 2016, Vol. 466, No. 3, pp. 276–281.

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Belov, A.A., Kalitkin, N.N. & Poshivaylo, I.P. Geometrically adaptive grids for stiff Cauchy problems. Dokl. Math. 93, 112–116 (2016). https://doi.org/10.1134/S1064562416010129

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  • DOI: https://doi.org/10.1134/S1064562416010129

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