The performance of phase-lag enhanced explicit Runge-Kutta Nyström pairs on \(N\)-body problems
Explicit Runge-Kutta Nyström methods (RKN\(p\)(\(q\)), where \(p\) is the order of the formulae to propagate the solution, and \(q\) is the order of the formulae to estimate the local error of integration) with enhanced phase-lag order (i.e., with free parameters selected to ensure the highest possible order for the integration of \(\ddot{y}=-\omega^2 y\) with \(\omega\) constant) are compared to general purpose RNK\(p\)(\(q\)) methods for long-time integrations of \(N\)-body problems. The aim of the authors is to see gains in the efficiency for long-time integrations, attending to the fact that the solutions of systems describing the motion of planets or satellites are someway close to oscillatory solutions of an harmonic oscillator. In the paper, the emphasis is made on the RKN8(6) pairs suitable for long-time integrations, but RKN4(3) and RKN6(4) have also been included just to gain insight in what way the efficiency depends on the order. Also integrations have been performed and compared using double and quadruple precision. The \(N\)-body problems are considered, and some of the main results achieved are the following: \begin{itemize}\item[1)] For Kepler's two-body problem used as a basic reference problem, it is detected that the pairs with enhanced phase-lag order are usually more efficient than the general purpose pairs for low eccentricities, and less efficient for high eccentricities. \item[2)] For the motion of the Sun and the planets Jupiter, Saturn, Uranus and Neptune the authors show that the enhanced RKN8(6) pair is more efficient than the general RKN8(6) pair for local error tolerances greater than 10\(^{-11}\), and less efficient for more severe ones. \item[3)] The same question as before, but including the motion of the Hellin-Roman-Crocket comet which has close encounters with Jupiter. Here the enhanced RKN8(6) appears to be slightly more efficient for lax accuracy requirements, and of similar efficiency otherwise. \item[4)] Finally, for the orbital motion of seven satellites of Saturn the enhanced RKN8(6) pair is more efficient than the general RKN8(6) pair.\end{itemize}
scientific article; zbMATH DE number 6015751
scientific article