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A trigonometrically fitted explicit Numerov-type method for second-order initial value problems with oscillating solutions. (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:
65L06Multistep, Runge-Kutta, and extrapolation methods
65L05Initial value problems for ODE (numerical methods)
34A34Nonlinear ODE and systems, general
65L20Stability and convergence of numerical methods for ODE
34C25Periodic solutions of ODE
34C10Qualitative theory of oscillations of ODE: zeros, disconjugacy and comparison theory