Runge-Kutta-Nyström methods adapted to the numerical integration of perturbed oscillators. (English) Zbl 1019.65050

The paper deals with methods for the numerical integration of perturbed oscillators, i.e.nonstiff initial value problems of the form \(y''(t) + \omega^2 y(t) = f(t, y(t), y'(t))\), \(y(t_0) = y_0\), \(y'(t_0) = y_0'\), where \(f\) is assumed to be small in magnitude. For this purpose, the author derives explicit Runge-Kutta-Nyström methods of order up to 5. The order conditions are discussed in detail. Numerical examples are provided showing the efficiency of the algorithms.


65L06 Multistep, Runge-Kutta and extrapolation methods for ordinary differential equations
65L20 Stability and convergence of numerical methods for ordinary differential equations
34A34 Nonlinear ordinary differential equations and systems
65L05 Numerical methods for initial value problems involving ordinary differential equations
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[1] Bettis, D. G., Numerical integration of products of Fourier and ordinary polynomials, Numer. Math., 14, 421-434 (1970) · Zbl 0198.49601
[2] Bettis, D. G., Runge-Kutta algorithms for oscillatory problems, J. Appl. Math. Phys. (ZAMP), 30, 699-704 (1979) · Zbl 0412.65038
[3] Franco, J. M.; Correas, J. M.; Pétriz, F., Métodos adaptados de tipo Störmer-Cowell de orden elevado, Revista Internacional de Métodos Numéricos para Cálculo y Diseño en Ingenierı́a, 7, 2, 193-216 (1991)
[4] Franco, J. M.; Palacián, J. F., High order adaptive methods of Nyström-Cowell type, J. Comput. Appl. Math., 81, 115-134 (1997) · Zbl 0879.65050
[5] González, A. B.; Martı́n, P.; Farto, J. M., A new family of Runge-Kutta type methods for the numerical integration of perturbed oscillators, Numer. Math., 82, 635-646 (1999) · Zbl 0935.65075
[6] Hairer, E.; Nørsett, S. P.; Wanner, S. P., Solving Ordinary Differential Equations I, Nonstiff Problems (1993), Springer-Verlag: Springer-Verlag Berlin · Zbl 0789.65048
[7] Jain, M. K., A modification of the Stiefel-Bettis method for nonlinear damped oscillators, BIT, 28, 302-307 (1988) · Zbl 0646.65063
[8] Martı́n, P.; Ferrándiz, J. M., Multistep numerical methods based on the Scheifele G-functions with applications to satellite dynamics, SIAM J. Numer. Anal., 34, 359-375 (1997) · Zbl 0878.65066
[9] Moore, P., Orbitally stable multistep methods, Celes. Mech., 17, 281-298 (1978) · Zbl 0388.65034
[10] Paternoster, B., Runge-Kutta(-Nyström) methods for ODEs with periodic solutions based on trigonometric polynomials, Appl. Numer. Math., 28, 401-412 (1998) · Zbl 0927.65097
[11] Sanz-Serna, J. M.; Calvo, M. P., Numerical Hamiltonian Problems (1994), Chapman and Hall: Chapman and Hall London · Zbl 0816.65042
[12] Stiefel, E.; Bettis, D. G., Stabilization of Cowell’s method, Numer. Math., 13, 154-175 (1969) · Zbl 0219.65062
[13] Stiefel, E. L.; Scheifele, G., Linear and Regular Celestial Mechanics (1971), Springer-Verlag: Springer-Verlag New York · Zbl 0226.70005
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