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High order explicit Runge–Kutta Nyström pairs

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Abstract

Explicit Runge–Kutta Nyström pairs provide an efficient way to find numerical solutions to second-order initial value problems when the derivative is cheap to evaluate. We present new optimal pairs of orders ten and twelve from existing families of pairs that are intended for accurate integrations in double precision arithmetic. We also present a summary of numerical comparisons between the new pairs on a set of eight problems which includes realistic models of the Solar System. Our searching for new order twelve pairs shows that there is often not quantitative agreement between the size of the principal error coefficients and the efficiency of the pairs for the tolerances we are interested in. Our numerical comparisons, as well as establishing the efficiency of the new pairs, show that the order ten pairs are more efficient than the order twelve pairs on some problems, even at limiting precision in double precision.

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Authors and Affiliations

  1. Department of Mathematics, University of Auckland, Private Bag 92019, Auckland, New Zealand

    Philip W. Sharp & Mohammad A. Qureshi

  2. The Jet Propulsion Laboratory, California Institute of Technology, Pasadena, CA, USA

    Kevin R. Grazier

Authors
  1. Philip W. Sharp
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  2. Mohammad A. Qureshi
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  3. Kevin R. Grazier
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Correspondence to Philip W. Sharp.

Additional information

The work of the second author was supported by the Higher Education Commission of Pakistan.

The work of the third author has been conducted in part at the Jet Propulsion Laboratory, California Institute of Technology under a contract with the National Aeronautics and Space Administration. Government sponsorship acknowledged.

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Sharp, P.W., Qureshi, M.A. & Grazier, K.R. High order explicit Runge–Kutta Nyström pairs. Numer Algor 62, 133–148 (2013). https://doi.org/10.1007/s11075-012-9571-0

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  • DOI: https://doi.org/10.1007/s11075-012-9571-0

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