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Stepsize selection for tolerance proportionality in explicit Runge–Kutta codes

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Abstract

The potential for adaptive explicit Runge–Kutta (ERK) codes to produce global errors that decrease linearly as a function of the error tolerance is studied. It is shown that this desirable property may not hold, in general, if the leading term of the locally computed error estimate passes through zero. However, it is also shown that certain methods are insensitive to a vanishing leading term. Moreover, a new stepchanging policy is introduced that, at negligible extra cost, ensures a robust global error behaviour. The results are supported by theoretical and numerical analysis on widely used formulas and test problems. Overall, the modified stepchanging strategy allows a strong guarantee to be attached to the complete numerical process.

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Authors and Affiliations

  1. Departamento de Matemática Aplicada, Universidad de Zaragoza, 50009, Zaragoza, Spain

    M. Calvo, J.I. Montijano & L. Rández

  2. Department of Mathematics, University of Strathclyde, Glasgow, G1 1XH, Scotland, UK

    D.J. Higham

Authors
  1. M. Calvo
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  2. D.J. Higham
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  3. J.I. Montijano
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  4. L. Rández
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Calvo, M., Higham, D., Montijano, J. et al. Stepsize selection for tolerance proportionality in explicit Runge–Kutta codes. Advances in Computational Mathematics 7, 361–382 (1997). https://doi.org/10.1023/A:1018959222223

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