On the zerostability of multistep methods on smooth nonuniform grids
(2018) In BIT Numerical Mathematics 58(4). p.11251143 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 BDFtype methods.
(Less)
 author
 Söderlind, Gustaf ^{LU} ; Fekete, Imre and Faragó, István
 organization
 publishing date
 20180710
 type
 Contribution to journal
 publication status
 published
 subject
 keywords
 BDF methods, Convergence, Initial value problems, Linear multistep methods, Nonuniform grids, Variable step size, Zero stability
 in
 BIT Numerical Mathematics
 volume
 58
 issue
 4
 pages
 1125  1143
 publisher
 Springer
 external identifiers

 scopus:85049665841
 ISSN
 00063835
 DOI
 10.1007/s105430180716y
 language
 English
 LU publication?
 yes
 id
 4ba6d12b11d044ff8350109c5c504489
 date added to LUP
 20180724 12:18:38
 date last changed
 20190220 11:22:56
@article{4ba6d12b11d044ff8350109c5c504489, 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 BDFtype methods.</p>}, author = {Söderlind, Gustaf and Fekete, Imre and Faragó, István}, issn = {00063835}, keyword = {BDF methods,Convergence,Initial value problems,Linear multistep methods,Nonuniform grids,Variable step size,Zero stability}, language = {eng}, month = {07}, number = {4}, pages = {11251143}, publisher = {Springer}, series = {BIT Numerical Mathematics}, title = {On the zerostability of multistep methods on smooth nonuniform grids}, url = {http://dx.doi.org/10.1007/s105430180716y}, volume = {58}, year = {2018}, }