2018/04/13 by Julien Alexandre dit Sandretto, Sandretto, Julien Alexandre dit
Computer Science · Mathematics · #34A45 #65G20 #65G40 #Classical Analysis and ODEs (math.CA) #Constraint Satisfaction and Optimization #FOS: Mathematics #Formal Methods in Verification #Numerical Analysis (math.NA) #Numerical Methods and Algorithms #cs.NA #math.CA #math.NA #msc:34A45 #msc:65G20 #msc:65G40
paper · pdf · doi:10.48550/arxiv.1804.04847
This is a revised version of "Runge-Kutta Theory and Constraint Programming", Reliable Computing vol. 25, 2017
arxiv created 2018/04/13 · openalex publication_date 2018/04/13 · arxiv updated 2018/04/16 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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.