Journal of Applied Mathematics and Decision Sciences
Volume 4 (2000), Issue 2, Pages 143-150
On order symplectic explicit Runge-Kutta Nyström methods
Department of Mathematics, University of Auckland, Auckland Private Bag 92019, New Zealand
Copyright © 2000 Lin-Yi Chou and P. W. Sharp. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
Order five symplectic explicit Runge-Kutta Nyström methods of
five stages are known to exist. However, these methods do not have
free parameters with which to minimise the principal error
coefficients. By adding one derivative evaluation per step, to give
either a six-stage non-FSAL family or a seven-stage FSAL family of
methods, two free parameters become available for the minimisation.
This raises the possibility of improving the efficiency of order five
methods despite the extra cost of taking a step.
We perform a minimisation of the two families to obtain an optimal
method and then compare its numerical performance with published
methods of orders four to seven. These comparisons along with those
based on the principal error coefficients show the new method is
significantly more efficient than the five-stage, order five methods.
The numerical comparisons also suggest the new methods can be more
efficient than published methods of other orders.