*(English)*Zbl 1136.65068

Summary: A new kind of trigonometrically fitted explicit Numerov-type method for the numerical integration of second-order initial value problems (IVPs) with oscillating or periodic solutions is presented. This new method is based on the original fifth-order method which is dispersion of order eight and dissipation of order five proposed by *J. M. Franco*, J. Comput. Appl. Math. 187, No. 1, 41–57 (2006; Zbl 1082.65071)].

Using trigonometrically fitting, we derive a more efficient method with higher accuracy for the numerical integration of second-order IVPs with oscillating solutions, and we analyze the stability, phase-lag(dispersion) and dissipation by the theory considered by *J. P. Coleman* and *L. Gr. Ixaru* [IMA J. Numer. Anal. 16, No. 2, 179–199 (1996; Zbl 0847.65052)].

Numerical experiments are carried out to show the efficiency of our new method in comparison with the methods proposed by *J. M. Franco* [J. Comput. Appl. Math. 187, No. 1, 41–57 (2006; Zbl 1060.65073)] and the exponentially fitted fourth order Runge-Kutta-Nyström method derived by *J. M. Franco* [J. Comput. Appl. Math. 167, No. 1, 1–19 (2004; Zbl 1060.65073)].

##### MSC:

65L06 | Multistep, Runge-Kutta, and extrapolation methods |

65L05 | Initial value problems for ODE (numerical methods) |

34A34 | Nonlinear ODE and systems, general |

65L20 | Stability and convergence of numerical methods for ODE |

34C25 | Periodic solutions of ODE |

34C10 | Qualitative theory of oscillations of ODE: zeros, disconjugacy and comparison theory |