High-Order Explicit Runge-Kutta Methods

A number of new explicit high-order Runge-Kutta methods have recently been discovered by Dr. Terry Feagin.  The methods are known as m-symmetric methods.  The concept of m-symmetry greatly simplifies the generation of high-order methods with reasonable numbers of stages. The 10th-order method requires 17 stages, the 12th-order requires 25 stages, and the 14th-order method requires 35 stages.  The approach is similar to that used by Hairer in developing a 10th-order method in 1978.

The coefficients:

The following text files contain the coefficients of the respective methods:

A numerical experiment comparing RK12(10) with an Extrapolation Method*

on the Pleiades problem (a two-dimensional celestial mechanics problem with seven bodies involving quasi-collisions for which automatic step size control is essential).

For more details on this problem, see:

E. Hairer, S.P. Norsett, and G. Wanner. Solving Ordinary Differential Equations I: Nonstiff Problems. Springer-Verlag, second revised edition, 1993

-Log10(error) plotted versus  Log10 of the number of derivative function evaluations required (NF)

* This method is the default extrapolation method used in the function NDSolve in Mathematica

          (viz., the Explicit Modified Midpoint Method with Gragg smoothing step as described in:

W. B. Gragg, "On extrapolation algorithms for ordinary initial value problems". SIAM J. Num. Anal., 2, 1965, 384-403.)