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doi: 10.1145/114697.103150pmid: N/A
To augment the discrete Runge-Kutta solutlon to the mitlal value problem, piecewlse Hermite interpolants have been used to provide a continuous approximation with a continuous first derivative We show that it M possible to construct mterpolants with arbltrardy many continuous derivatives which have the same asymptotic accuracy and basic cost as the Hermite interpol ants. We also show that the usual truncation coefficient analysis can be applied to these new interpolants, allowing their accuracy to be examined in more detad As an Illustration, we present some globally C2 interpolants for use with a popular 4th and 5th order Runge-Kutta pair of Dormand and Prince, and we compare them theoretically and numerically with existing interpolants.
doi: 10.1145/114697.116811pmid: N/A
We performed numerical testing of six explicit Runge-Kutta pairs ranging in order from a (3,4) pair to a (7,8) pair. All the test problems had smooth solutions and we assumed dense output was not required. The pairs were implemented in a uniform way. In particular, the stepsize selection for all pairs was based on the locally optimal formula. We tested the efficiency of the pairs, to what extent tolerance proportionality held, the accuracy of the local error estimate and stepsize prediction, and the performance on mildly stiff problems. We also showed, for these pairs, how the performance could be altered noticeably by making simple changes to the stepsize selection strategy. As part of the work, we demonstrated new ways of presenting numerical comparisons. <?Pub Fmt italic>—From the Author's Abstract<?Pub Fmt /italic>
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