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A four-step phase-fitted method for the numerical integration of second order initial-value problems. (English) Zbl 0726.65089

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: ${y}^{\text{'}\text{'}}\left(x\right)=f\left(x,y\right),\phantom{\rule{1.em}{0ex}}y\left({x}_{0}\right)={y}_{0},\phantom{\rule{1.em}{0ex}}{y}^{\text{'}}\left({x}_{0}\right)={y}_{0}^{\text{'}}·$ Examples occur in celestial mechanics, in quantum mechanical scattering problems and elsewhere.

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.

The first is the “almost periodic” problem studied by E. Stiefel and D. G. Bettis [Numer. Math. 13, 154-175 (1969; Zbl 0219.65062)]: ${z}^{\text{'}\text{'}}+z=0·001{e}^{ix},\phantom{\rule{1.em}{0ex}}z\left(0\right)=1,\phantom{\rule{1.em}{0ex}}{z}^{\text{'}}\left(0\right)=0·9995i,\phantom{\rule{1.em}{0ex}}z\in C$ and the other is the resonance problem of the one-dimensional Schrödinger equation: ${y}^{\text{'}\text{'}}\left(x\right)=f\left(x\right)y\left(x\right),$ $x\in \left[0,\infty \right)$, with $f\left(x\right)=W\left(x\right)-E,$ $W\left(x\right)=\ell \left(\ell +1\right)/{x}^{2}+V\left(x\right),\ell \in ℤ$, E is the energy (E$\in ℝ\right)$. In both problems the new suggested method is more accurate than other methods with minimal phase-lag, especially for large step-sizes.

##### MSC:
 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
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