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GRID-INDEPENDENT CONSTRUCTION OF MULTISTEP METHODS

Arévalo, Carmen LU and Söderlind, Gustaf LU (2017) In Journal of Computational Mathematics 35. p.672-692
Abstract
A new polynomial formulation of variable step size linear multistep methods is presented, where each k-step method is characterized by a fixed set of k-1 or k parameters. This construction includes all methods of maximal order (p=k for stiff, and p=k+1 for nonstiff problems). Supporting time step adaptivity by construction, the new formulation is not based on extending classical fixed step size methods; instead classical methods are obtained as fixed step size restrictions within a unified framework. The methods are implemented in Matlab, with local error estimation and a wide range of step size controllers. This provides a platform for investigating and comparing different multistep method in realistic operational conditions.... (More)
A new polynomial formulation of variable step size linear multistep methods is presented, where each k-step method is characterized by a fixed set of k-1 or k parameters. This construction includes all methods of maximal order (p=k for stiff, and p=k+1 for nonstiff problems). Supporting time step adaptivity by construction, the new formulation is not based on extending classical fixed step size methods; instead classical methods are obtained as fixed step size restrictions within a unified framework. The methods are implemented in Matlab, with local error estimation and a wide range of step size controllers. This provides a platform for investigating and comparing different multistep method in realistic operational conditions. Computational experiments show that the new multistep method construction and implementation compares favorably to existing software, although variable order has not yet been included. (Less)
Please use this url to cite or link to this publication:
author
organization
publishing date
type
Contribution to journal
publication status
published
subject
keywords
Linear multistep methods, Variable step size, Adaptive step size, Step size control, Explicit methods, Implicit methods, Nonstiff methods, Stiff methods, Initial value problems, Ordinary differential equations, Differential-algebraic equations, Implementation
in
Journal of Computational Mathematics
volume
35
pages
672 - 692
publisher
Global Science Press
ISSN
0254-9409
DOI
10.4208/jcm.1611-m2015-0404
language
English
LU publication?
yes
id
8f52415d-7359-4ca5-9d68-c78648590f08
alternative location
http://www.global-sci.org/jcm/readabs.php?vol=35&no=5&page=672&year=2017&ppage=692
date added to LUP
2017-08-24 14:42:41
date last changed
2017-11-06 12:21:39
@article{8f52415d-7359-4ca5-9d68-c78648590f08,
  abstract     = {A new polynomial formulation of variable step size linear multistep methods is presented, where each k-step method is characterized by a fixed set of k-1 or k parameters. This construction includes all methods of maximal order (p=k for stiff, and p=k+1 for nonstiff problems). Supporting time step adaptivity by construction, the new formulation is not based on extending classical fixed step size methods; instead classical methods are obtained as fixed step size restrictions within a unified framework. The methods are implemented in Matlab, with local error estimation and a wide range of step size controllers. This provides a platform for investigating and comparing different multistep method in realistic operational conditions. Computational experiments show that the new multistep method construction and implementation compares favorably to existing software, although variable order has not yet been included.},
  author       = {Arévalo, Carmen and Söderlind, Gustaf},
  issn         = {0254-9409},
  keyword      = {Linear multistep methods,Variable step size,Adaptive step size,Step size control,Explicit methods,Implicit methods, Nonstiff methods,Stiff methods,Initial value problems,Ordinary differential equations,Differential-algebraic equations,Implementation},
  language     = {eng},
  pages        = {672--692},
  publisher    = {Global Science Press},
  series       = {Journal of Computational Mathematics},
  title        = {GRID-INDEPENDENT CONSTRUCTION OF MULTISTEP METHODS},
  url          = {http://dx.doi.org/10.4208/jcm.1611-m2015-0404},
  volume       = {35},
  year         = {2017},
}