Embedded implicit Runge–Kutta Nyström method for solving second-order differential equations

Abstract

An embedded diagonally implicit Runge–Kutta Nyström (RKN) method is constructed for the integration of initial-value problems for second-order ordinary differential equations possessing oscillatory solutions. This embedded method is derived using a three-stage diagonally implicit RKN method of order four within which a third-order three stage diagonally implicit RKN method is embedded. We demonstrate how this system can be solved, and by an appropriate choice of free parameters, we obtain an optimized RKN(4,3) embedded algorithm. We also examine the intervals of stability and show that the method is strongly stable within an appropriate region of stability and is thus suitable for oscillatory problems by applying the method to the test equation y″=−ω² y, ω>0. Necessary and sufficient conditions are given for this method to possess non-vanishing intervals of periodicity, for the fourth-order method. Finally, we present the coefficients of the method optimized for small truncation errors. This new scheme is likely to be efficient for the numerical integration of second-order differential equations with periodic solutions, using adaptive step size.

Keywords:

• Initial value problems
• Diagonally implicit
• Second-order equations
• Embedded Runge–Kutta Nyström
• Step-size control

Acknowledgements

The partial support of the Third World Academy of Sciences (TWAS), Italy and Council of Scientific and Industrial Research (CSIR), India is gratefully acknowledged. We would also like to thank Dr. Gangan Prathap, SIC CSIR-Centre for Mathematical Modeling and Computer Simulation, Bangalore, for his support and encouragement. The advice and insight provided by P.W Sharp, J.M Fine, B.P Sommeijer and M.E.A Mikkawy is also gratefully acknowledged.