  There exist many Runge-Kutta methods (explicit or implicit), more or less
adapted to specific problems. Some of them have interesting properties, such as
stability for stiff problems or symplectic capability for problems with energy
conservation. Defining a new method suitable to a given problem has become a
challenge. The size, the complexity and the order do not stop growing. This
informal challenge to implement the best method is interesting but an important
unsolved problem persists. Indeed, the coefficients of Runge-Kutta methods are
harder and harder to compute, and the result is often expressed in
floating-point numbers, which may lead to erroneous integration schemes. Here,
we propose to use interval analysis tools to compute Runge-Kutta coefficients.
In particular, we use a solver based on guaranteed constraint programming.
Moreover, with a global optimization process and a well chosen cost function,
we propose a way to define some novel optimal Runge-Kutta methods.
