D. J. Higham
Abstract
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.
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Index Terms
- Highly continuous Runge-Kutta interpolants
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Reviews
Peter Bruce Worland
Sometimes one needs to provide output from the numerical solution of initial value problems in ordinary differential equations that is more dense than one would normally obtain from the discretized approximation. This paper extends the work of Shampine [1], who derived Hermite-type interpolants to Runge-Kutta approximations with global C 1 continuity. Higham shows that these interpolants may be adapted to provide arbitrary smoothness (in practice, with global C 2 continuity) with competitive accuracy and cost. The author presents a general derivation of the interpolants and a rigorous analysis of the truncation error. Specific examples are obtained with fifth- and sixth-order local accuracies and global C 2 continuity. A detailed analysis determines practical limits imposed on the range of stepsize changes in adaptive codes in order to maintain acceptable levels of error in the interpolants. The author does a credible job of defending his choices. For examples, he points out the deficiencies in the alternative strategy of employing a cubic spline interpolant to the output from the Runge-Kutta global approximation. Reading Shampine [1] and this paper will provide a good overview of the general problem and current approaches to its solution.
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