https://hal.science/hal-01762196Alexandre Dit Sandretto, JulienJulienAlexandre Dit SandrettoU2IS - Unité d'Informatique et d'Ingénierie des Systèmes - ENSTA Paris - École Nationale Supérieure de Techniques AvancéesValidated Runge-Kutta Methods for Initial Value ProblemsHAL CCSD2018[MATH.MATH-DS] Mathematics [math]/Dynamical Systems [math.DS]Alexandre dit Sandretto, Julien2018-04-09 17:12:592022-05-11 15:20:032018-04-20 13:43:18enConference papers1Validated numerical integration methods based on explicit or 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 methods is the main contribution of this talk. More precisely, an efficient solution to the challenge of making validated Runge-Kutta methods is presented. We also present a new interval contractor approach to solve implicit Runge-Kutta methods. We also propose, in this talk, to use interval analysis tools to compute Runge-Kutta coefficients, in particular, 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.