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

Gustaf Söderlind, Imre Fekete, István Faragó

Research output: Contribution to journalArticle

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 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.

Original languageEnglish
Pages (from-to)1-19
Number of pages19
JournalBIT Numerical Mathematics
DOIs
Publication statusAccepted/In press - Jul 10 2018

Fingerprint

Non-uniform Grid
Multistep Methods
Zero
Step-size Control
Grid
Linear multistep Methods
Equidistant
Error function
Tolerance
Adjacent

Keywords

  • BDF methods
  • Convergence
  • Initial value problems
  • Linear multistep methods
  • Nonuniform grids
  • Variable step size
  • Zero stability

ASJC Scopus subject areas

  • Software
  • Computer Networks and Communications
  • Computational Mathematics
  • Applied Mathematics

Cite this

On the zero-stability of multistep methods on smooth nonuniform grids. / Söderlind, Gustaf; Fekete, Imre; Faragó, István.

In: BIT Numerical Mathematics, 10.07.2018, p. 1-19.

Research output: Contribution to journalArticle

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