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Zbl 0726.65089
Raptis, A.D.; Simos, T.E.
A four-step phase-fitted method for the numerical integration of second order initial-value problems. (English)
[J] BIT 31, No.1, 160-168 (1991). ISSN 0006-3835; ISSN 1572-9125/e

A four-step method with phase-lag of infinite order is developed for the numerical integration of second order initial-value problems of the form: $y''(x)=f(x,y),\quad y(x\sb 0)=y\sb 0,\quad y'(x\sb 0)=y'\sb 0.$ Examples occur in celestial mechanics, in quantum mechanical scattering problems and elsewhere. \par The idea is to maintain a free parameter $\alpha$ in the method such that the method to be fitted to an oscillatory component of the theoretical solution. Applications of the new method have been done in two problems. \par The first is the ``almost periodic'' problem studied by {\it E. Stiefel} and {\it D. G. Bettis} [Numer. Math. 13, 154-175 (1969; Zbl 0219.65062)]: $z''+z=0.001e\sp{ix},\quad z(0)=1,\quad z'(0)=0.9995i,\quad z\in C$ and the other is the resonance problem of the one-dimensional Schrödinger equation: $y''(x)=f(x)y(x),$ $x\in [0,\infty)$, with $f(x)=W(x)-E,$ $W(x)=\ell (\ell +1)/x\sp 2+V(x),\ell \in {\bbfZ}$, E is the energy (E$\in {\bbfR})$. In both problems the new suggested method is more accurate than other methods with minimal phase-lag, especially for large step-sizes.
[M.Gousidou-Koutita (Thessaloniki)]

MSC 2000:: *65L06 Multistep, Runge-Kutta, and extrapolation methods
65L05 Initial value problems for ODE (numerical methods)
34A34 Nonlinear ODE and systems, general
34C25 Periodic solutions of ODE
34L40 Particular ordinary differential operators

Keywords: phase-fitted method; four-step method; phase-lag; second order initial- value problems; ``almost periodic'' problem; resonance problem; Schrödinger equation

Citations: Zbl 0219.65062

Cited in: Zbl 1117.65109

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