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The several methods that are not identi ed in Figure 3.5.2 are the more traditional
formulas, including the classical RungeKutta method (solid line). All of these are listed
in Hairer et al. 16], Section II.1. \Fehlberg's method" is the fourth and fthorder pair
given in Example 3.5.4. The \DormandPrince" method is the fourth and fthorder
pair of Example 3.5.5. \Butcher's method" is the sixthorder sevenstage formula shown
in Table 3.5.1. It is the only formula that is beyond fourth or fth order. Results in the
lower graph of Figure 3.5.2 use local extrapolation thus, the higherorder solution of the
pair is kept, even though it has no local error estimate.
Of all the methods shown in Figure 3.5.2, the DormandPrince and Fehlberg methods
appear to have the greatest accuracy for a given cost. The higherorder Butcher formula
gains appeal as accuracy increases. The DormandPrince method has a distinct advantage
relative to the Fehlberg method when local extrapolation is used. As noted, the Dorm...
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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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