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

Norazak Senu,Mohamed Suleiman,Norihan Md Arifin,Fudziah Ismail

doi:10.1155/2012/324989

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.

Highlights

In many scientific areas of engineering and applied sciences such as celestial mechanics, quantum mechanics, elastodynamics, theoretical physics and chemistry, and electronics, oscillatory problems of second-order ordinary differential equations ODEs can be found
We evaluate the effectiveness of the new DIRKN pairs derived in the previous section when they are applied to the numerical solution of eight problems which are model and nonmodel problems for constant and variable step size
The new DIRKN4 3 6New method is more accurate compared with RKND method while comparable with PFRKN method for certain problem

Summary

Introduction

In many scientific areas of engineering and applied sciences such as celestial mechanics, quantum mechanics, elastodynamics, theoretical physics and chemistry, and electronics, oscillatory problems of second-order ordinary differential equations ODEs can be found. An m-stage Runge-Kutta-Nystrom RKN method for the numerical integration of the IVP is given by m yn 1 yn hyn h2 bif tn cih, Yi i1 m yn 1 yn h bif tn cih, Yi , i1

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