Runge-Kutta-Nystr\"om symplectic splitting methods of order 8

TL;DR

This paper introduces new eighth-order Runge-Kutta-Nyström symplectic splitting methods that outperform existing methods in efficiency for medium to high accuracy integrations of second-order ODEs.

Contribution

The paper develops and tests new eighth-order RKN symplectic splitting methods, demonstrating improved efficiency over existing symmetric compositions and lower-order methods.

Findings

Better efficiency than symmetric compositions of second-order schemes

Outperforms RKN methods of orders 4 and 6 at medium to high accuracy

More efficient than extrapolation methods for certain high-accuracy, short-interval problems

Abstract

Different families of Runge-Kutta-Nystr\"om (RKN) symplectic splitting methods of order 8 are presented for second-order systems of ordinary differential equations and are tested on numerical examples. They show a better efficiency than state-of-the-art symmetric compositions of 2nd-order symmetric schemes and RKN splitting methods of orders 4 and 6 for medium to high accuracy. For some particular examples, they are even more efficient than extrapolation methods for high accuracies and integrations over relatively short time intervals.