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On the zero-stability of multistep methods on smooth nonuniform grids

Söderlind, Gustaf LU ; Fekete, Imre and Faragó, István (2018) In BIT Numerical Mathematics
Abstract

In order to be convergent, linear multistep methods must be zero stable. While constant step size theory was established in the 1950’s, zero stability on nonuniform grids is less well understood. Here we investigate zero stability on compact intervals and smooth nonuniform grids. In practical computations, step size control can be implemented using smooth (small) step size changes. The resulting grid (Formula presented.) can be modeled as the image of an equidistant grid under a smooth deformation map, i.e., (Formula presented.), where (Formula presented.) and the map (Formula presented.) is monotonically increasing with (Formula presented.) and (Formula presented.). The model is justified for any fixed order method operating in its... (More)

In order to be convergent, linear multistep methods must be zero stable. While constant step size theory was established in the 1950’s, zero stability on nonuniform grids is less well understood. Here we investigate zero stability on compact intervals and smooth nonuniform grids. In practical computations, step size control can be implemented using smooth (small) step size changes. The resulting grid (Formula presented.) can be modeled as the image of an equidistant grid under a smooth deformation map, i.e., (Formula presented.), where (Formula presented.) and the map (Formula presented.) is monotonically increasing with (Formula presented.) and (Formula presented.). The model is justified for any fixed order method operating in its asymptotic regime when applied to smooth problems, since the step size is then determined by the (smooth) principal error function which determines (Formula presented.), and a tolerance requirement which determines N. Given any strongly stable multistep method, there is an (Formula presented.) such that the method is zero stable for (Formula presented.), provided that (Formula presented.). Thus zero stability holds on all nonuniform grids such that adjacent step sizes satisfy (Formula presented.) as (Formula presented.). The results are exemplified for BDF-type methods.

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author
organization
publishing date
type
Contribution to journal
publication status
epub
subject
keywords
BDF methods, Convergence, Initial value problems, Linear multistep methods, Nonuniform grids, Variable step size, Zero stability
in
BIT Numerical Mathematics
pages
19 pages
publisher
Springer
external identifiers
  • scopus:85049665841
ISSN
0006-3835
DOI
10.1007/s10543-018-0716-y
language
English
LU publication?
yes
id
4ba6d12b-11d0-44ff-8350-109c5c504489
date added to LUP
2018-07-24 12:18:38
date last changed
2018-07-24 12:18:38
@article{4ba6d12b-11d0-44ff-8350-109c5c504489,
  abstract     = {<p>In order to be convergent, linear multistep methods must be zero stable. While constant step size theory was established in the 1950’s, zero stability on nonuniform grids is less well understood. Here we investigate zero stability on compact intervals and smooth nonuniform grids. In practical computations, step size control can be implemented using smooth (small) step size changes. The resulting grid (Formula presented.) can be modeled as the image of an equidistant grid under a smooth deformation map, i.e., (Formula presented.), where (Formula presented.) and the map (Formula presented.) is monotonically increasing with (Formula presented.) and (Formula presented.). The model is justified for any fixed order method operating in its asymptotic regime when applied to smooth problems, since the step size is then determined by the (smooth) principal error function which determines (Formula presented.), and a tolerance requirement which determines N. Given any strongly stable multistep method, there is an (Formula presented.) such that the method is zero stable for (Formula presented.), provided that (Formula presented.). Thus zero stability holds on all nonuniform grids such that adjacent step sizes satisfy (Formula presented.) as (Formula presented.). The results are exemplified for BDF-type methods.</p>},
  author       = {Söderlind, Gustaf and Fekete, Imre and Faragó, István},
  issn         = {0006-3835},
  keyword      = {BDF methods,Convergence,Initial value problems,Linear multistep methods,Nonuniform grids,Variable step size,Zero stability},
  language     = {eng},
  month        = {07},
  pages        = {19},
  publisher    = {Springer},
  series       = {BIT Numerical Mathematics},
  title        = {On the zero-stability of multistep methods on smooth nonuniform grids},
  url          = {http://dx.doi.org/10.1007/s10543-018-0716-y},
  year         = {2018},
}