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Families of three-stage third order Runge-Kutta-Nyström methods for y″ = f (x, y, y′)

Published online by Cambridge University Press:  17 February 2009

M. M. Chawla  and
S. R. Sharma
M. M. Chawla
Affiliation:
Department of Mathematics, Indian Institute of Technology, Hauz Khas, New Delhi-110016, India.
S. R. Sharma
Affiliation:
Department of Mathematics, Deshbandhu College, Kalkaji, New Dethi-110019, India.
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Abstract

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In this paper all three-stage third order (explicit) Runge-Kutta-Nyström (R-K-N) methods for y″ = f (x, y, y′) are presented. While determining particular methods we require that when these methods are applied to the test equation: y″ − (α + β) y ′ + αβy = 0, the measure of the relative error F, introduced by Rutishauser [4], should not deteriorate in the case of equal eigenvalues (β → α). Further, we require that when these methods are applied to special differential equations y″ = f(x, y) they should possess either of the two properties: (P1) a method remains of order three but is two-stage, (P2) a method remains three-stage but attains order four. We present new R-K-N methods which are stabilized in the sense of Ruthishauser [4] and which possess the property (P1). (There does not exist any three-stage third order R-K-N method which is stabilized and which possesses the property (P2).)

Type
Research Article
Information
The ANZIAM Journal , Volume 26 , Issue 3 , January 1985 , pp. 375 - 386
Copyright
Copyright © Australian Mathematical Society 1985

References

[1]Butcher, J. C., “On the attainable order of Runge-Kutta methods”, Math. Comp. 19 (1965), 408–417.CrossRefGoogle Scholar
[2]Hairer, E. and Wanner, G., “A theory for Nystrōm methods”, Numer. Math. 25 (1976), 383–400.CrossRefGoogle Scholar
[3]Henrici, P., Discrete variable methods in ordinary differential equations (John Wiley, New York,1962).Google Scholar
[4]Rutishauser, H., “Bemerkungen zur numerischen Integration gewohnhicher Differential gleichungen n-ter Ordnung”, Numer. Math. 2 (1960), 263–279.CrossRefGoogle Scholar