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)].