Electronic Journal of Differential Equations,
Conference 02 (1999), pp. 1-9.

Title: Fifth-order Runge-Kutta with higher order derivative approximations

Authors: David Goeken (The Univ. of Houston, TX, USA)
 Olin Johnson (The Univ. of Houston, TX, USA)
 
Abstract: 
Given $y'=f(y)$, standard Runge-Kutta
methods perform multiple evaluations of $f(y)$ in each integration
sub-interval as required for a given accuracy.  Evaluations of
$y''=f_{y}f$ or higher derivatives are not considered due
to the assumption that the calculations involved in these functions
exceed those of $f$.  However, $y''$ can be approximated to
sufficient accuracy from past and current evaluations of $f$ to
achieve a higher order of accuracy than is available through current functional
evaluations alone.
In July of 1998 at the ANODE (Auckland Numerical Ordinary Differential
Equations) Workshop, we introduced
a new class of Runge-Kutta methods based on this observation (Goeken~1999).
We presented a third-order method which requires
only two evaluations of $f$ and a fourth-order method which requires three.
This paper reviews these two methods and gives the general solution
to the equations generated by the fifth-order methods of this new class.
Interestingly, these fifth-order methods require only four functional
evaluations per step whereas standard Runge-Kutta methods require six.


Published December 9, 1999.
Math Subject Classifications: 65L06.
Key Words: Multistep Runge-Kutta; third-order method;
 fourth-order method; fifth-order method; higher order derivatives.