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An explicit hybrid method of Numerov type for second-order periodic initial-value problems. (English) Zbl 0844.65061
The author introduces and tests a new algorithm for approximation to periodic solutions of a nonlinearly perturbed system of linear ordinary differential equations having the form $$y''(t)= Ay(t)+ g(t, y(t)),\quad t_0\le t< \infty.\tag i$$ The algorithm is fourth-order four stage of Numerov type, and is designed to have minimal frequency distortion when $g\equiv 0$ [cf. {\it M. M. Chawla} and {\it P. S. Rao}, J. Comput. Appl. Math. 15, 329-337 (1986; Zbl 0598.65054)]. Numerical results obtained with four test problems indicate that the new method performs better than other methods of Numerov type as well as methods using symplectic integrations and the LSODE code.

##### MSC:
 65L06 Multistep, Runge-Kutta, and extrapolation methods 65L05 Initial value problems for ODE (numerical methods) 34C25 Periodic solutions of ODE 34A34 Nonlinear ODE and systems, general
ODEPACK
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##### References:
 [1] Calvo, M. P.: Canonical Runge-Kutta-Nyström methods. Ph.d. thesis (1992) · Zbl 0802.65089 [2] Chawla, M. M.: Numerov made explicit has better stability. Bit 24, 117-118 (1984) · Zbl 0568.65042 [3] Chawla, M. M.; Rao, P. S.: A numerov-type method with minimal phase-lag for the integration of second order periodic initial-value problems. J. comput. Appl. math. 11, 277-281 (1984) · Zbl 0565.65041 [4] Chawla, M. M.; Rao, P. S.: A numerov-type method with minimal phase-lag for the integration of second order periodic initial-value problems. II: explicit method. J. comput. Appl. math. 15, 329-337 (1986) · Zbl 0598.65054 [5] Franco, J. M.; Palacios, M.: High-order P-stable multistep methods. J. comput. Appl. math. 30, 1-10 (1990) · Zbl 0726.65091 [6] Gladwell, I.; Thomas, R. M.: Damping and phase analysis for some methods for solving second order ordinary differential equations. Internat. J. Numer. methods eng. 19, 495-503 (1983) · Zbl 0513.65053 [7] Hindmarsh, A. C.: Odepack, a systematized collection of ODE solvers. Scientific computing, 55-64 (1983) [8] Kinoshita, H.; Yoshida, H.; Nakai, H.: Simplectic integrators and their application to dynamical astronomy. Celest. mech. Dyn. astron. 50, 59-71 (1991) · Zbl 0724.70019 [9] Lambert, J. D.; Watson, I. A.: Symmetric multistep methods for periodic initial value problems. J. ima 18, 189-202 (1976) · Zbl 0359.65060 [10] Meneguette, M.: Chawla-numerov method revisited. J. comput. Appl. math. 36, 247-250 (1991) · Zbl 0751.65048 [11] Sanz-Serna, J. M.: Symplectic integrators for Hamiltonian problems: an overview. Acta numerica 1, 243-286 (1992) · Zbl 0762.65043 [12] Van Der Houwen, P. J.; Sommeijer, B. P.: Explicit Runge-Kutta (-Nyström) methods with reduced phase errors for computing oscillating solutions. SIAM J. Numer. anal. 24, 595-617 (1987) · Zbl 0624.65058 [13] Yoshida, H.: Construction of higher order symplectic integrators. Phys. lett. A 150, 262-268 (1990)