On order5symplectic explicit Runge-Kutta Nyström methods

[摘要] Order five symplectic explicit Runge-Kutta Nyström methods offive stages are known to exist. However, these methods do not havefree parameters with which to minimise the principal errorcoefficients. By adding one derivative evaluation per step, to giveeither a six-stage non-FSAL family or a seven-stage FSAL family ofmethods, two free parameters become available for the minimisation.This raises the possibility of improving the efficiency of order fivemethods despite the extra cost of taking a step.We perform a minimisation of the two families to obtain an optimalmethod and then compare its numerical performance with publishedmethods of orders four to seven. These comparisons along with thosebased on the principal error coefficients show the new method issignificantly more efficient than the five-stage, order five methods.The numerical comparisons also suggest the new methods can be moreefficient than published methods of other orders.

[学科分类] 自然科学（综合）