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 C1 continuity. Higham shows that these interpolants may be adapted to provide arbitrary smoothness (in practice, with global C2 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 C2 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.
