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Predictor-corrector methods for periodic second-order initial-value problems. (English) Zbl 0631.65074

This paper deals with the numerical solution of initial value problems for the special second order equation ${y}^{\text{'}\text{'}}=f\left(t,y\right)$, by means of predictor-corrector methods. The authors define the phase lag with respect to the homogeneous solution component as the phase error introduced by the numerical scheme when applied to the test equation ${y}^{\text{'}\text{'}}=-{k}^{2}y$. Thus some predictor-corrector methods introduced by the authors [ibid. 3, 417-437 (1983; Zbl 0533.65045)] for first order differential equations are modified to make them applicable to special second order equations and the coefficients are chosen satisfying the requirement that both the algebraic and phase lag orders be as large as possible.

In this way several numerical predictor-corrector schemes with orders 4 and 6 and phase lag orders up to 10 are proposed. The paper ends with some numerical experiments comparing the behaviour of these methods and the classical fourth order Runge-Kutta-Nyström (RKN) method. It is concluded that for problems with periodic solutions the proposed schemes are more efficient than the RKN method.

Reviewer: M.Calvo
##### MSC:
 65L05 Initial value problems for ODE (numerical methods) 34A34 Nonlinear ODE and systems, general 34C25 Periodic solutions of ODE