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 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,
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.)