A trigonometrically fitted explicit Numerov-type method for second-order initial value problems with oscillating solutions Original Research Article

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 in this paper. This new method is based on the original fifth-order method which is dispersion of order eight and dissipation of order five proposed in [J.M. Franco, A class of explicit two-step hybrid methods for second-order IVPs, J. Comput. Appl. Math. 187 (2006) 41–57]. 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 Coleman and Ixaru (see [J.P. Coleman, L.Gr. Ixaru, P-stability and exponential-fitting methods for y″=f(x,y)y″=f(x,y), IMA J. Numer. Anal. 16 (1996) 179–199]). Numerical experiments are carried out to show the efficiency of our new method in comparison with the methods proposed in [J.M. Franco, A class of explicit two-step hybrid methods for second-order IVPs, J. Comput. Appl. Math. 187 (2006) 41–57] and the exponentially fitted fourth order RKN method derived in [J.M. Franco, Exponentially fitted explicit Runge–Kutta–Nyström methods, J. Comput. Appl. Math. 167 (2004) 1–19].