×

zbMATH — the first resource for mathematics

Runge-Kutta methods with minimum error bounds. (English) Zbl 0105.31903
L’A. étudie l’erreur sur un pas dans la méthode Runge-Kutta pour une équation du 1er ordre, sous l’hypothèse \(\partial^{i+j}f/\partial x^i\partial y^j < L^{i+j}/M^{j-1}\). Il retrouve pour l’ordre 2 la méthode classique de Heun \((\alpha_2 = \frac23)\), pour l’ordre 3 une méthode \((\alpha_2 = \frac12\), \(\alpha_3 = \frac34)\), pour l’ordre 4 \((\alpha_2 = \frac25\), \(\alpha_3\approx 0,45)\) et aussi \((\alpha_2 = \frac25\), \(\alpha_3 = \frac35\). L’A. qui ne cite, en dehors du mémoire de Kutta que de la littérature en langue anglaise, paraît ignorer diverses publications où le problème de la minimisation est abordé par des méthodes analogues aux siennes, entre autres celles du rapporteur [Chiffres, Revue Assoc. franç. Calcul 2, 21–26 (1959; Zbl 0094.11302)].
Reviewer: J. Kuntzmann

MSC:
65L06 Multistep, Runge-Kutta and extrapolation methods for ordinary differential equations
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] Ludwig Bieberbach, Theorie der Differentialgleichungen, Dover Publications, New York, 1944 (German). · Zbl 0063.00379
[2] John W. Carr III, Error bounds for the Runge-Kutta single-step integration process, J. Assoc. Comput. Mach. 5 (1958), 39 – 44. · Zbl 0087.32503
[3] B. A. Galler and D. P. Rozenberg, A generalization of a theorem of Carr on error bounds for Runga-Kutta procedures, J. Assoc. Comput. Mach. 7 (1960), 57 – 60. · Zbl 0096.10101
[4] S. Gill, A process for the step-by-step integration of differential equations in an automatic digital computing machine, Proc. Cambridge Philos. Soc. 47 (1951), 96 – 108. · Zbl 0042.13202
[5] Zdeněk Kopal, Numerical analysis. With emphasis on the application of numerical techniques to problems of infinitesimal calculus in single variable, John Wiley & Sons, Inc., New York, 1955. · Zbl 0065.10702
[6] W. Kutta, “Beitrag zur näherungsweiser Integration totaler Differentialgleichungen,” Z. Math. Phys., v. 46, 1901, p. 435-453. · JFM 32.0316.02
[7] Max Lotkin, On the accuracy of Runge-Kutta’s method, Math. Tables and Other Aids to Computation 5 (1951), 128 – 133. · Zbl 0044.33104
[8] W. E. Milne, Note on the Runge-Kutta method, J. Research Nat. Bur. Standards 44 (1950), 549 – 550.
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.