Generation of High-Order Interpolants for Explicit Runge-Kutta Pairs

Abstract

Explicit Runge-Kutta pairs can be enhanced by providing them with interpolants. Enhancements include the ability to estimate and control the defect, to produce dense output, and to calculate past values in delay differential equations. The coefficients of an interpolant are easily generated by bootstrapping on the order conditions. However, the generation of high-order interpolants requires a large number of arithmetic operations. We describe an efficient algorithm for the generation of high-order interpolants and illustrate the use of the algorithm with three applications.