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Digital filters in adaptive time-stepping

Söderlind, Gustaf LU (2003) In ACM Transactions on Mathematical Software 29(1). p.1-26
Abstract
Adaptive time-stepping based on linear digital control theory has several advantages: the algorithms can be analyzed in terms of stability and adaptivity, and they can be designed to produce smoother stepsize sequences resulting in significantly improved regularity and computational stability. Here, we extend this approach by viewing the closed-loop transfer map H-(φ) over cap: log (φ) over cap bar right arrow log h as a digital filter, processing the signal log (φ) over cap ( the principal error function) in the frequency domain, in order to produce a smooth stepsize sequence log h. The theory covers all previously considered control structures and offers new possibilities to construct stepsize selection algorithms in the asymptotic... (More)
Adaptive time-stepping based on linear digital control theory has several advantages: the algorithms can be analyzed in terms of stability and adaptivity, and they can be designed to produce smoother stepsize sequences resulting in significantly improved regularity and computational stability. Here, we extend this approach by viewing the closed-loop transfer map H-(φ) over cap: log (φ) over cap bar right arrow log h as a digital filter, processing the signal log (φ) over cap ( the principal error function) in the frequency domain, in order to produce a smooth stepsize sequence log h. The theory covers all previously considered control structures and offers new possibilities to construct stepsize selection algorithms in the asymptotic stepsize-error regime. Without incurring extra computational costs, the controllers can be designed for special purposes such as higher order of adaptivity ( for smooth ODE problems) or a stronger ability to suppress high-frequency error components (nonsmooth problems, stochastic ODEs). Simulations verify the controllers' ability to produce stepsize sequences resulting in improved regularity and computational stability. (Less)
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author
organization
publishing date
type
Contribution to journal
publication status
published
subject
keywords
error control, digital filters, control theory, algorithm analysis, adaptivity, algorithms, theory, stepsize control, mathematical software
in
ACM Transactions on Mathematical Software
volume
29
issue
1
pages
1 - 26
publisher
ACM
external identifiers
  • wos:000181815300001
  • scopus:3042616322
ISSN
0098-3500
DOI
10.1145/641876.641877
language
English
LU publication?
yes
id
c552045a-a6fd-43dd-ae4a-15f950b09b1a (old id 315626)
date added to LUP
2007-09-22 11:30:46
date last changed
2018-01-07 09:18:48
@article{c552045a-a6fd-43dd-ae4a-15f950b09b1a,
  abstract     = {Adaptive time-stepping based on linear digital control theory has several advantages: the algorithms can be analyzed in terms of stability and adaptivity, and they can be designed to produce smoother stepsize sequences resulting in significantly improved regularity and computational stability. Here, we extend this approach by viewing the closed-loop transfer map H-(φ) over cap: log (φ) over cap bar right arrow log h as a digital filter, processing the signal log (φ) over cap ( the principal error function) in the frequency domain, in order to produce a smooth stepsize sequence log h. The theory covers all previously considered control structures and offers new possibilities to construct stepsize selection algorithms in the asymptotic stepsize-error regime. Without incurring extra computational costs, the controllers can be designed for special purposes such as higher order of adaptivity ( for smooth ODE problems) or a stronger ability to suppress high-frequency error components (nonsmooth problems, stochastic ODEs). Simulations verify the controllers' ability to produce stepsize sequences resulting in improved regularity and computational stability.},
  author       = {Söderlind, Gustaf},
  issn         = {0098-3500},
  keyword      = {error control,digital filters,control theory,algorithm analysis,adaptivity,algorithms,theory,stepsize control,mathematical software},
  language     = {eng},
  number       = {1},
  pages        = {1--26},
  publisher    = {ACM},
  series       = {ACM Transactions on Mathematical Software},
  title        = {Digital filters in adaptive time-stepping},
  url          = {http://dx.doi.org/10.1145/641876.641877},
  volume       = {29},
  year         = {2003},
}