Validated Explicit and Implicit Runge-Kutta Methods∗†

A set of validated numerical integration methods based on explicit and implicit Runge-Kutta schemes is presented to solve, in a guaranteed way, initial value problems of ordinary differential equations. Runge-Kutta methods are well-known to have strong stability properties, which make them appealing to be the basis of validated numerical integration methods. A new approach to bound the local truncation error of any Runge-Kutta method is the main contribution of this article, which pushes back the current state of the art. More precisely, an efficient solution to the challenge of making validated Runge-Kutta methods is presented, based on the theory of John Butcher. We also present a new interval contractor approach to solve implicit Runge-Kutta methods. A complete experimentation based on Vericomp benchmark is described.