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On the error control in ODE solvers with local extrapolation

Zur Fehlerkontrolle in Differentialgleichungs-Codes mit lokaler Extrapolation

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Abstract

Jackson, Enright, and Hull have analytically assessed the effectiveness of RK-algorithms for linear ODEs with constant coefficients. Their results may carry over to more general problems if a certain relation between the local error estimate and the true local error of the algorithm continues to hold. Numerical investigation of this relation for a code which performs local extrapolation gives some insight in the performance of the error control mechanism of such codes.

Zusammenfassung

Jackson, Enright und Hull haben auf analytischem Weg die Effektivität von RK-Algorithmen für lineare Differentialgleichungen mit konstanten Koeffizienten bewertet. Ihre Ergebnisse könnten sich auf allgemeinere Probleme übertragen lassen, falls eine gewisse Beziehung zwischen dem geschätzten und dem wahren lokalen Fehler weiter gilt. Die numerische Untersuchung dieses Sachverhalts für einen Code mit “lokaler Extrapolation” gibt einen Einblick in die Wirkungsweise der Fehlerkontrolle bei solchen Codes.

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References

  1. Jackson, K. R., Enright, W. H., Hull, T. E.: A theoretical criterion for comparing Runge-Kutta formulas. SINUM15, 618–641 (1978).

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  2. Enright, W. H.: Using a testing package for the automatic assessment of numerical methods for ODE's, in: Performance Evaluation of Numerical Software (Fosdick, Ll., ed.), pp. 199–213. North-Holland 1979.

  3. Weinmüller, E.: Erweiterung des Toronto-Modells zur Effizienzuntersuchung von numerischen Verfahren für gewöhnliche Differentialgleichungen, Bericht Nr. 45/80, Inst. f. Num. Math. TU Wien, 1980.

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Authors and Affiliations

  1. Institut für Numerische Mathematik, Technische Universität Wien, Gusshausstrasse 27-29, A-1040, Wien, Austria

    H. J. Stetter & E. Weinmüller

Authors
  1. H. J. Stetter
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  2. E. Weinmüller
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Stetter, H.J., Weinmüller, E. On the error control in ODE solvers with local extrapolation. Computing 27, 169–177 (1981). https://doi.org/10.1007/BF02243551

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  • DOI: https://doi.org/10.1007/BF02243551

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