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Constant coefficient linear multistep methods with step density control

Arévalo, Carmen LU ; Söderlind, Gustaf LU and Diaz, José LU (2007) In Journal of Computational and Applied Mathematics 205(2). p.891-900
Abstract
In linear multistep methods with variable step size, the method's coefficients are functions of the step size ratios. The coefficients therefore need to be recomputed on every step to retain the method's proper order of convergence. An alternative approach is to use step density control to make the method adaptive. If the step size sequence is smooth, the method can use constant coefficients without losing its order of convergence. The paper introduces this new adaptive technique and demonstrates its feasibility with a few test problems. The technique works in perfect agreement with theory for a given step density function. For practical use, however, the density must be generated with data computed from the numerical solution. We... (More)
In linear multistep methods with variable step size, the method's coefficients are functions of the step size ratios. The coefficients therefore need to be recomputed on every step to retain the method's proper order of convergence. An alternative approach is to use step density control to make the method adaptive. If the step size sequence is smooth, the method can use constant coefficients without losing its order of convergence. The paper introduces this new adaptive technique and demonstrates its feasibility with a few test problems. The technique works in perfect agreement with theory for a given step density function. For practical use, however, the density must be generated with data computed from the numerical solution. We introduce a local error tracking controller, which automatically adapts the density to computed data, and demonstrate in computational experiments that the technique works well at least up to fourth-order methods. (c) 2006 Elsevier B.V. All rights reserved. (Less)
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author
organization
publishing date
type
Contribution to journal
publication status
published
subject
keywords
density control, non-uniform grid, step, step size selection, multistep method, variable step size method
in
Journal of Computational and Applied Mathematics
volume
205
issue
2
pages
891 - 900
publisher
Elsevier
external identifiers
  • wos:000247261300021
  • scopus:34248153725
ISSN
0377-0427
DOI
10.1016/j.cam.2006.02.060
language
English
LU publication?
yes
id
0930bb11-377e-45ff-942e-4992b23d7942 (old id 648811)
date added to LUP
2007-12-04 15:25:11
date last changed
2017-02-13 13:10:02
@article{0930bb11-377e-45ff-942e-4992b23d7942,
  abstract     = {In linear multistep methods with variable step size, the method's coefficients are functions of the step size ratios. The coefficients therefore need to be recomputed on every step to retain the method's proper order of convergence. An alternative approach is to use step density control to make the method adaptive. If the step size sequence is smooth, the method can use constant coefficients without losing its order of convergence. The paper introduces this new adaptive technique and demonstrates its feasibility with a few test problems. The technique works in perfect agreement with theory for a given step density function. For practical use, however, the density must be generated with data computed from the numerical solution. We introduce a local error tracking controller, which automatically adapts the density to computed data, and demonstrate in computational experiments that the technique works well at least up to fourth-order methods. (c) 2006 Elsevier B.V. All rights reserved.},
  author       = {Arévalo, Carmen and Söderlind, Gustaf and Diaz, José},
  issn         = {0377-0427},
  keyword      = {density control,non-uniform grid,step,step size selection,multistep method,variable step size method},
  language     = {eng},
  number       = {2},
  pages        = {891--900},
  publisher    = {Elsevier},
  series       = {Journal of Computational and Applied Mathematics},
  title        = {Constant coefficient linear multistep methods with step density control},
  url          = {http://dx.doi.org/10.1016/j.cam.2006.02.060},
  volume       = {205},
  year         = {2007},
}