Local error estimation and step size control in adaptive linear multistep methods
(2021) In Numerical Algorithms 86(2). p.537-563- Abstract
In a k-step adaptive linear multistep methods the coefficients depend on the k − 1 most recent step size ratios. In a similar way, both the actual and the estimated local error will depend on these step ratios. The classical error model has been the asymptotic model, chp+ 1y(p+ 1)(t), based on the constant step size analysis, where all past step sizes simultaneously go to zero. This does not reflect actual computations with multistep methods, where the step size control selects the next step, based on error information from previously accepted steps and the recent step size history. In variable step size implementations the error model must therefore be dynamic and include past step ratios, even in the asymptotic... (More)
In a k-step adaptive linear multistep methods the coefficients depend on the k − 1 most recent step size ratios. In a similar way, both the actual and the estimated local error will depend on these step ratios. The classical error model has been the asymptotic model, chp+ 1y(p+ 1)(t), based on the constant step size analysis, where all past step sizes simultaneously go to zero. This does not reflect actual computations with multistep methods, where the step size control selects the next step, based on error information from previously accepted steps and the recent step size history. In variable step size implementations the error model must therefore be dynamic and include past step ratios, even in the asymptotic regime. In this paper we derive dynamic asymptotic models of the local error and its estimator, and show how to use dynamically compensated step size controllers that keep the asymptotic local error near a prescribed tolerance tol. The new error models enable the use of controllers with enhanced stability, producing more regular step size sequences. Numerical examples illustrate the impact of dynamically compensated control, and that the proper choice of error estimator affects efficiency.
(Less)
- author
- Arévalo, Carmen LU ; Söderlind, Gustaf LU ; Hadjimichael, Yiannis and Fekete, Imre
- organization
- publishing date
- 2021
- type
- Contribution to journal
- publication status
- published
- subject
- keywords
- Adaptivity, Control theory, Differential equations, Dynamic compensator, Initial value problems, Linear multistep methods, Local error estimation, Step size control, Time stepping, Variable step size
- in
- Numerical Algorithms
- volume
- 86
- issue
- 2
- pages
- 537 - 563
- publisher
- Springer
- external identifiers
-
- scopus:85086134014
- ISSN
- 1017-1398
- DOI
- 10.1007/s11075-020-00900-1
- language
- English
- LU publication?
- yes
- id
- d46182f0-0fbe-4fc0-91de-57ca79391ad6
- date added to LUP
- 2020-07-03 11:47:36
- date last changed
- 2022-04-18 23:15:21
@article{d46182f0-0fbe-4fc0-91de-57ca79391ad6,
abstract = {{<p>In a k-step adaptive linear multistep methods the coefficients depend on the k − 1 most recent step size ratios. In a similar way, both the actual and the estimated local error will depend on these step ratios. The classical error model has been the asymptotic model, ch<sup>p+ 1</sup>y<sup>(p+ 1)</sup>(t), based on the constant step size analysis, where all past step sizes simultaneously go to zero. This does not reflect actual computations with multistep methods, where the step size control selects the next step, based on error information from previously accepted steps and the recent step size history. In variable step size implementations the error model must therefore be dynamic and include past step ratios, even in the asymptotic regime. In this paper we derive dynamic asymptotic models of the local error and its estimator, and show how to use dynamically compensated step size controllers that keep the asymptotic local error near a prescribed tolerance tol. The new error models enable the use of controllers with enhanced stability, producing more regular step size sequences. Numerical examples illustrate the impact of dynamically compensated control, and that the proper choice of error estimator affects efficiency.</p>}},
author = {{Arévalo, Carmen and Söderlind, Gustaf and Hadjimichael, Yiannis and Fekete, Imre}},
issn = {{1017-1398}},
keywords = {{Adaptivity; Control theory; Differential equations; Dynamic compensator; Initial value problems; Linear multistep methods; Local error estimation; Step size control; Time stepping; Variable step size}},
language = {{eng}},
number = {{2}},
pages = {{537--563}},
publisher = {{Springer}},
series = {{Numerical Algorithms}},
title = {{Local error estimation and step size control in adaptive linear multistep methods}},
url = {{http://dx.doi.org/10.1007/s11075-020-00900-1}},
doi = {{10.1007/s11075-020-00900-1}},
volume = {{86}},
year = {{2021}},
}