Journal of Applied Mathematics and Decision Sciences
Volume 4 (2000), Issue 2, Pages 143-150

On order 5 symplectic explicit Runge-Kutta Nyström methods

Lin-Yi Chou and P. W. Sharp

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.