Continuous Runge-Kutta-Nyström methods for initial value problems with periodic solutions

In the present work, we are concerned with the derivation of continuous Runge-Kutta-Nyström methods for the numerical treatment of second-order ordinary differential equations with Nyström methods for the numerical treatment of second-order ordinary differential equations with periodic solutions. Numerical methods used for solving such problems are better to have the characteristics of high phase-lag order. First we analyse the construction algorithm for a high phase-lag order scaled extension of an explicit Runge-Kutta-Nyström method. Using this procedure, we manage to construct a phase-lag order 14 continuous extension of a popular nine stages 8(6) order ERKN pair. In the literature, only phase-lag order 12 continuous extension of nine stage 8(6) ERKN pairs can be found, so the proposed scaling method has the higher, until now, dispersion order. Numerical tests for the proposed methods are done over various test problems.