High order adaptive methods of Nyström-Cowell type. (English) Zbl 0879.65050

A class of unconditionally stable multistep methods for second-order periodic or quasiperiodic initial value problems is developed. Problems of this type occur very frequently in celestial mechanics, nonlinear oscillations and in other situations. The methods belong to the category of methods with nonconstant coefficients. The coefficients of the methods depend upon the “frequency” of the problem, i.e. upon a parameter \(\omega > 0\). The methods integrate exactly trigonometric functions and algebraic polynomials. It is known that these methods for \(\omega = 0\) are reduced to classical methods, i.e. methods with constant coefficients.
In this paper adaptive Nyström-Cowell methods of arbitrarily high order of accuracy are constructed. These methods are reduced to classical Nyström-Cowell methods for \(\omega = 0\). Numerical illustrations indicate the efficiency of the new methods.
Reviewer: T.E.Simos (Xanthi)


65L06 Multistep, Runge-Kutta and extrapolation methods for ordinary differential equations
34A34 Nonlinear ordinary differential equations and systems
34C25 Periodic solutions to ordinary differential equations
65L05 Numerical methods for initial value problems involving ordinary differential equations
65L20 Stability and convergence of numerical methods for ordinary differential equations
Full Text: DOI


[1] Abad, A.; Deprit, A.; Elipe, A.; Sein-Echaluce, M., A perturbed elliptic oscillator: flow inversion through butterfly bifurcations, Rev. Acad. Cienc. Zaragoza, 44, 89-105 (1989)
[2] Correas, J. M., Linear Multistep Schemes based on algebraic, trigonometric and exponential polynomials, (Proceedings of the First World Conference on Mathematics at the Service of Man, vol. 1 (1977), Universidad Politécnica de Barcelona: Universidad Politécnica de Barcelona Spain), 339-359
[3] Correas, J. M.; Martín, M. C., Métodos multipaso de coefficientes variables para el P.V.I. 2 especial, (V-Jornadas Mat. Luso-Españolas, 3 (1978), Universidad de Aveiro: Universidad de Aveiro Portugal), 940-962
[4] Franco, J. M.; Correas, J. M., Métodos de tipo Störmer-Cowell con coefficientes variables. Applicationes a la integración de movimientos orbitales, (Actas del X-CEDYA, 1 (1987), Universidad Politécnica de Valencia), 117-124
[5] Henrici, P., Discrete variable methods in ODE’s (1962), Wiley: Wiley New York · Zbl 0112.34901
[6] Jain, M. K., P-stable methods for periodic I.V.P. of second order differential equations, BIT, 19, 347-355 (1979) · Zbl 0495.65033
[7] Kirchgraber, U., An ODE-solver based on the method of averaging, Numer. Math., 53, 621-652 (1988) · Zbl 0656.65072
[8] Neta, B.; Ford, C. H., Families of methods for ordinary differential equations based on trigonometric polynomials, J. Comput. Appl. Math., 10, 33-38 (1984) · Zbl 0529.65050
[9] Palacios, M.; Franco, J. M., Adaptive Cowell methods for orbital motions, (Teles, J., Orbital Mechanics and Mission Design. Orbital Mechanics and Mission Design, Proceedings of AAS/NASA, vol. 69 (1989), AAS Publications Office: AAS Publications Office San Diego, California 92128), 3-22
[10] Prince, P. J.; Dormand, J. R., High order embedded Runge-Kutta formulae, J. Comput. Appl. Math., 7, 65-75 (1981) · Zbl 0449.65048
[11] Stiefel, E.; Bettis, D. G., Stabilization of Cowell’s methods, Numer. Math., 13, 154-175 (1969) · Zbl 0219.65062
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.