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**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.

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 |

### Keywords:

implicit Runge-Kutta-Nyström methods; predictor-corrector methods; direct methods; second-order differential equations; numerical comparison; performance; convergence; numerical examples### Citations:

Zbl 0731.65071
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\textit{N. h. Cong}, J. Comput. Appl. Math. 45, No. 3, 347--355 (1993; Zbl 0774.65047)

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### References:

[1] | Fehlberg, E., Klassische Runge—Kutta—Nyström Formeln mit Schrittweiten-Kontrolle für Differentialgleichungen x″ = f(t, x), Computing, 10, 305-315 (1972) · Zbl 0261.65046 |

[2] | Hairer, E.; Nørsett, S. P.; Wanner, G., Solving Ordinary Differential Equations, I. Nonstiff Problems (1987), Springer: Springer Berlin · Zbl 0638.65058 |

[4] | huu Cong, Nguyen, Note on the performance of direct collocation-based parallel-iterated Runge—Kutta—Nyström methods, Report NM-R9214 (1992), Centre Math. Comput. Sci: Centre Math. Comput. Sci Amsterdam · Zbl 0940.65514 |

[6] | van der Houwen, P. J.; Sommeijer, B. P.; huu Cong, Nguyen, Stability of collocation-based Runge—Kutta—Nyström methods, BIT, 31, 469-481 (1991) · Zbl 0731.65071 |

[7] | van der Houwen, P. J.; Sommeijer, B. P.; huu Cong, Nguyen, Parallel diagonally implicit Runge—Kutta—Nyström methods, Appl. Numer. Math., 9, 2, 111-131 (1992) · Zbl 0747.65059 |

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