Validated Explicit and Implicit Runge-Kutta Methods

Julien Alexandre Dit Sandretto (1) , Alexandre Chapoutot (1)

1 SFL - Sûreté et Fiabilité des Logiciels

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 methods 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.

Mots clés: Differential equations; Validated simulation

Domaines: Modélisation et simulation; Systèmes dynamiques [math.DS]

Julien Alexandre Dit Sandretto, Alexandre Chapoutot. Validated Explicit and Implicit Runge-Kutta Methods. Reliable Computing electronic edition, 2016, Special issue devoted to material presented at SWIM 2015, 22. ⟨hal-01243053⟩