David Goeken & Olin Johnson
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_yf 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 November 23, 1999.
Subject lassfications: 65L06
Key words: multistep Runge-Kutta, third-order method,
fourth-order method, fifth-order method, higher order derivatives.
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David Goeken Department of Computer Science The University of Houston Houston, TX 77204-3475, USA e-mail: dgoeken@cs.uh.edu Now with the LinCom Corporation, Houston, Texas |
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Olin Johnson Department of Computer Science The University of Houston Houston, TX 77204-3475, USA e-mail: johnson@cs.uh.edu |
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