Interpolants for Runge-Kutta formulas

A general procedure for the construction of interpolants for Runge-Kutta (RK) formulas is presented. As illustrations, this approach is used to develop interpolants for three explicit RK formulas, including those employed in the well-known subroutines RKF45 and DVERK. A typical result is that no extra function evaluations are required to obtain an interpolant with O ( h 5 ) local truncation error for the fifth-order RK formula used in RKF45; two extra function evaluations per step are required to obtain an interpolant with O ( h 6 ) local truncation error for this RK formula.