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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)

MSC:

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
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