Collocation Runge–Kutta–Nyström methods for solving second-order initial value problems

ABSTRACT

A class of modified collocation Runge–Kutta–Nyström (RKN) methods for solving second-order initial value problem $y^{″}=f(t,y,y^{\prime })$, $y(t_0)=y_0$, $y^{\prime }(t_0)=y_0^{\prime }$ has been formulated and studied in the paper. The new methods are applicable to a larger class of second-order initial value problems compared to the classical collocation RKN methods and they reduce to the classical methods when $y^{\prime }$ is absent from the equation. Superconvergence for the new methods is attained when the set of collocation points satisfies orthogonality conditions. We proved that an s-stage modified collocation RKN method is of accuracy order of at least s for any set of collocation parameters $\mathit{c}$ and at most $2s$ when $\mathit{c}$ are Gauss points. The stability function and the stability of some modified RKN methods have also been investigated. Numerical experiments are included to demonstrate the advantage of the new methods.

Keywords:

• Runge–Kutta–Nyström
• superconvergence
• collocation
• second-order IVP

2010 Mathematics Subject Classifications:

• 65L05
• 65L06
• 65L20
• 65L60

Acknowledgments

The author thanks the referees for their useful comments that improved the paper.

Disclosure statement

No potential conflict of interest was reported by the author(s).