## Note on the performance of direct and indirect Runge-Kutta-Nyström methods.(English)Zbl 0774.65047

This paper is concerned with the numerical solution of special second- order ordinary differential equations by means of implicit Runge-Kutta- Nyström (RKN) methods. Two kinds of RKN methods are considered by the author. Firstly, the so called indirect methods which can be derived from implicit methods for first-order differential equations. Secondly, the direct methods which are constructed for special second-order differential equations [see e.g., the author, P. J. van der Houwen and B. P. Someijer, BIT 31, No. 3, 469-481 (1991; Zbl 0731.65071)].
The aim of the paper is to carry out a numerical comparison of the performance of both types of methods (with the same order) when they are implemented in a predictor-corrector mode and applied to (non stiff) linear second-order differential equations. It turns out that for the direct methods the convergence factors and error constants are smaller than those of their corresponding indirect methods. Moreover two numerical examples are presented to show the superiority of the direct over the indirect methods.
Reviewer: M.Calvo (Zaragoza)

### MSC:

 65L06 Multistep, Runge-Kutta and extrapolation methods for ordinary differential equations 65L05 Numerical methods for initial value problems involving ordinary differential equations 34A34 Nonlinear ordinary differential equations and systems 65Y20 Complexity and performance of numerical algorithms

Zbl 0731.65071
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### References:

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