Validated computation of the local truncation error of Runge-Kutta methods with automatic differentiation

摘要

A novel approach to bound the local truncation error of explicit and implicit Runge-Kutta methods is presented. This approach takes its roots in the modern theory of Runge-Kutta methods, namely the order condition theorem, defined by John Butcher in the 1960s. More precisely, our work is an instance, for Runge-Kutta methods, of the generic algorithm defined by Ferenc Bartha and Hans Munthe-Kaas in 2014 which computes B-series with automatic differentiation techniques. In particular, this specialized algorithm is combined with set-membership framework to define validated numerical integration methods based on Runge-Kutta methods.