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Diagonally implicit Runge-Kutta-Nyström methods for oscillatory problems. (English) Zbl 0676.65072
A number of second order two and three stage diagonally implicit Runge- Kutta-Nyström (DIRKN) methods with low phase error are developed for the initial value problem $\left(i\right)\phantom{\rule{1.em}{0ex}}{y}^{\text{'}\text{'}}=f\left(t,y\right),$ $\left(ii\right)\phantom{\rule{1.em}{0ex}}y\left(0\right)={y}_{0},$ ${y}^{\text{'}}\left(0\right)={y}_{0}^{\text{'}}·$ The definition of phase error and amplitude error for the Runge-Kutta-Nyström algorithm associated with (i), (ii) has been developed by the authors in an earlier paper [ibid. 24, 595-617 (1987; Zbl 0624.65058)]. The phase error $\phi$ ($\omega$ h) and amplification error $\alpha$ ($\omega$ h) are defined as extensions of the explicitly known formulas for deviation of the phase and amplitude of the RKN solution of (i), (ii) from the true solution when $f=-{\omega }^{2}y$. An RKN method is said to be dispersive and dissipative of order q and r if $\left(iii\right)\phantom{\rule{1.em}{0ex}}\phi \left(\omega h\right)=O\left({h}^{q+1}\right),$ $\left(iv\right)\phantom{\rule{1.em}{0ex}}\alpha \left(\omega h\right)=O\left({h}^{r+1}\right)·$ The authors construct two-stage second order DIRKN algorithms in which $q=4,6$, one of which is unconditionally strongly stable. Also three-stage algorithms are constructed with $q=6,8,10$, one of which is P-stable. Results of computation of actual dispersion error are presented using the new algorithms on five test problems. The new algorithms are better than standard DIRKN methods in case the dispersion error grows larger than the truncation error.
Reviewer: J.B.Butler jun.
##### MSC:
 65L05 Initial value problems for ODE (numerical methods) 34A34 Nonlinear ODE and systems, general 34C10 Qualitative theory of oscillations of ODE: zeros, disconjugacy and comparison theory