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

- Article in a journal -
 

Author(s)
Olivier Mullier , Alexandre Chapoutot , dit Sandretto , Julien Alexandre

Published in
Special issue of Optimization Methods & Software: Advances in Algorithmic Differentiation Optimization Methods & Software

Editor(s)
Bruce Christianson, Shaun A. Forth, Andreas Griewank

Year
2018

Publisher
Taylor & Francis

Abstract
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.

Cross-References
Christianson2018Sio

BibTeX
@ARTICLE{
         Mullier2018Vco,
       crossref = "Christianson2018Sio",
       author = "Olivier Mullier and Alexandre Chapoutot and dit Sandretto, Julien Alexandre",
       title = "Validated computation of the local truncation error of {R}unge--{K}utta methods with
         automatic differentiation",
       journal = "Optimization Methods \& Software",
       volume = "33",
       number = "4--6",
       pages = "718--728",
       year = "2018",
       publisher = "Taylor \& Francis",
       doi = "10.1080/10556788.2018.1459620",
       url = "https://doi.org/10.1080/10556788.2018.1459620",
       eprint = "https://doi.org/10.1080/10556788.2018.1459620",
       abstract = "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.",
       booktitle = "Special issue of Optimization Methods \& Software: Advances in
         Algorithmic Differentiation",
       editor = "Bruce Christianson and Shaun A. Forth and Andreas Griewank"
}


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