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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 \[ y''(t)= Ay(t)+ g(t, y(t)),\quad t_0\leq t< \infty.\tag{i} \] The algorithm is fourth-order four stage of Numerov type, and is designed to have minimal frequency distortion when \(g\equiv 0\) [cf. M. M. Chawla and 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 for ordinary differential equations
65L05 Numerical methods for initial value problems involving ordinary differential equations
34C25 Periodic solutions to ordinary differential equations
34A34 Nonlinear ordinary differential equations and systems

Citations:

Zbl 0598.65054

Software:

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References:

[1] Calvo, M. P., Canonical Runge-Kutta-Nyström methods, (Ph.D. Thesis (1992), Universidad de Valladolid) · 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
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[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)
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