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Two-stage and three-stage diagonally implicit Runge-Kutta Nyström methods of orders three and four. (English) Zbl 0711.65057

The special second order initial value problem, ${y}^{\text{'}\text{'}}=f\left(x,y\right)$, $y\left({x}_{0}\right)={y}_{0}$, ${y}^{\text{'}}\left({x}_{0}\right)={y}_{0}^{\text{'}}$, can be efficiently treated by Runge- Kutta-Nyström (RKN) methods. Problems which are linear in y have motivated several definitions of stability of numerical methods. A method is R-stable if the amplitude of the numerical solution to ${y}^{\text{'}\text{'}}=-{\omega }^{2}y$ does not increase with x for all $\omega$. Such methods must be implicit, and of these, diagonally implicit methods are efficient to implement. The authors construct two- and three-stage methods of orders 3 and 4 and show which of them are R-stable. (NB. (3.2a) should read $1+{d}_{2}{\phi }^{2}\le 1·\right)·$

Further, they determine which methods (if any) are P-stable in which case there is no dissipation, and which are RL-stable in which case, oscillation is dissipated for large $\omega$. In addition, the dispersive order which needs to be high for accurate estimation of the phase of oscillatory components, is examined for these methods. The paper concludes with some particular methods, and some comments for their implementation.

Reviewer: J.H.Verner
##### MSC:
 65L06 Multistep, Runge-Kutta, and extrapolation methods 65L20 Stability and convergence of numerical methods for ODE 65L05 Initial value problems for ODE (numerical methods) 34A34 Nonlinear ODE and systems, general