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Discrete Dynamics in Nature and Society
Volume 2012 (2012), Article ID 324989, 20 pages
http://dx.doi.org/10.1155/2012/324989
Research Article

New 4(3) Pairs Diagonally Implicit Runge-Kutta-Nyström Method for Periodic IVPs

Department of Mathematics and Institute for Mathematical Research, Universiti Putra Malaysia, 43400 Serdang, Selangor, Malaysia

Received 14 February 2012; Revised 24 June 2012; Accepted 8 July 2012

Academic Editor: Taher S. Hassan

Copyright © 2012 Norazak Senu et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

New 4(3) pairs Diagonally Implicit Runge-Kutta-Nyström (DIRKN) methods with reduced phase-lag are developed for the integration of initial value problems for second-order ordinary differential equations possessing oscillating solutions. Two DIRKN pairs which are three- and four-stage with high order of dispersion embedded with the third-order formula for the estimation of the local truncation error. These new methods are more efficient when compared with current methods of similar type and with the L-stable Runge-Kutta pair derived by Butcher and Chen (2000) for the numerical integration of second-order differential equations with periodic solutions.