Runge-Kutta Theory and Constraint Programming

There exist many Runge-Kutta methods (explicit or implicit), more or less adapted to a given class of problems. Some of them have interesting properties such as A-stability for stiff problems or symplecticity for problems with energy conservation. Defining a new method, adapted to a given class of problems, has become a challenge. Indeed, the number of stages and the order don’t stop to increase. This race to the “best” method is interesting but forgot an important problem. More precisely, the coefficients of a Runge-Kutta method are more and more difficult to compute and the result is often given in floating-point numbers, which may lead to violate their definition rules. We propose a method using interval analysis tools to compute Runge-Kutta coefficients by using 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.