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Unformatted text preview: however, we must verify that this approximation provides an improved order of accuracy.
After all, Euler's method has an O(h2) local error and not an O(h3) error. Let's try to
verify that the combined scheme does indeed have an O(h3) local error by considering
the slightly more general scheme yn = yn 1 + h(b1 k1 + b2 k2) (3.1.5a) k1 = f (tn 1 yn 1) (3.1.5b) k2 = f (tn 1 + ch yn 1 + hak1 ): (3.1.5c) ; where
; ; ; ; Schemes of this form are an example of RungeKutta methods. We see that the proposed
midpoint scheme is recovered by selecting b1 = 0, b2 = 1, c = 1=2, and a = 1=2. We
also see that the method does not require any partial derivatives of f (t y). Instead, (the
potential) highorder accuracy is obtained by evaluating f (t y) at an additional time.
The coe cients a, b1 , b2 , and c will be determined so that a Taylor's series expansion
of (3.1.5) using the exact ODE solution matches the Taylor's series expansion (3.1.2,
3.1.3) of the exact ODE solution to as high a power in h as possible...
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This document was uploaded on 03/16/2014 for the course CSCI 6820 at Rensselaer Polytechnic Institute.
 Spring '14
 JosephE.Flaherty

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