New interpolants for runge-kutta algorithms using second derivatives

Abstract

In this paper a general interpolant for the Explicit Runge-Kutta methods is proposed. These interpolants are based on second derivatives on mesh-points of the integration interval, and first derivatives on interior points of each step. These first derivatives can be produced using lower order interpolants. Here an interpolant with 0(h ⁶) local truncation error for the fifth order solution used in RKF4(5) method is presented, with a cost of “about” one extra function evaluation per integration step.

Keywords:

• Ordinary differential equations
• initial value problem
• interpolant