Journal of Applied Mathematics
Volume 2012 (2012), Article ID 236281, 27 pages
http://dx.doi.org/10.1155/2012/236281
Research Article

A New Family of Phase-Fitted and Amplification-Fitted Runge-Kutta Type Methods for Oscillators

Zhaoxia Chen,1 Xiong You,1,2 Xin Shu,1,3 and Mei Zhang1

1Department of Applied Mathematics, Nanjing Agricultural University, Nanjing 210095, China
2State Key Laboratory for Novel Software Technology at Nanjing University, Nanjing 210093, China
3Department of Computer Science and Engineering, Shanghai Jiaotong University, Shanghai 200240, China

Received 19 April 2012; Revised 17 July 2012; Accepted 23 July 2012

Academic Editor: Jesus Vigo-Aguiar

Copyright © 2012 Zhaoxia Chen 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

In order to solve initial value problems of differential equations with oscillatory solutions, this paper improves traditional Runge-Kutta (RK) methods by introducing frequency-depending weights in the update. New practical RK integrators are obtained with the phase-fitting and amplification-fitting conditions and algebraic order conditions. Two of the new methods have updates that are also phase-fitted and amplification-fitted. The linear stability and phase properties of the new methods are examined. The results of numerical experiments on physical and biological problems show the robustness and competence of the new methods compared to some highly efficient integrators in the literature.