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Electronic Journal of Differential Equations
Electronic Journal of Differential Equations
15th annual Conference of Applied Mathematics, Univ. of Central Oklahoma,
Electron. J. Diff. Eqns., Conf. 02, 1999, pp. 19.
Fifthorder RungeKutta with higher order derivative approximations
David Goeken & Olin Johnson
Abstract:
Given y'=f(y), standard RungeKutta
methods perform multiple evaluations of f(y) in each integration
subinterval 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 RungeKutta methods based on this observation (Goeken 1999).
We presented a thirdorder method which requires
only two evaluations of f and a fourthorder method which requires three.
This paper reviews these two methods and gives the general solution
to the equations generated by the fifthorder methods of this new class.
Interestingly, these fifthorder methods require only four functional
evaluations per step whereas standard RungeKutta methods require six.
Published November 23, 1999.
Subject lassfications: 65L06
Key words: multistep RungeKutta, thirdorder method,
fourthorder method, fifthorder method, higher order derivatives.
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David Goeken
Department of Computer Science
The University of Houston
Houston, TX 772043475, USA
email: dgoeken@cs.uh.edu
Now with the LinCom Corporation, Houston, Texas


Olin Johnson
Department of Computer Science
The University of Houston
Houston, TX 772043475, USA
email: johnson@cs.uh.edu

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