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Analysis of Computational Algorithms for Linear Multistep Methods

Sjö, Anders LU (1999) In Doctoral Theses in Mathematical Sciences 1999:6.
Abstract
Linear multistep methods (LMMs) constitute a class of time-stepping methods for the solution of initial value ODEs; the most well-known methods of this class are the Adams methods (AMs) and the backward differentiation formulae (BDFs). For the fixed stepsize LMMs there exists an extensive error and stability analysis; in practical computations, however, one always uses a variable stepsize. It is well known that the varying stepsize affects the error and stability properties of the LMMs; the analysis of variable stepsize LMMs is generally much more complicated than for the fixed stepsize methods. Due to this, many of the computational algorithms in LMM codes are based on results from analyses for fixed stepsize LMMs or onestep methods, and... (More)
Linear multistep methods (LMMs) constitute a class of time-stepping methods for the solution of initial value ODEs; the most well-known methods of this class are the Adams methods (AMs) and the backward differentiation formulae (BDFs). For the fixed stepsize LMMs there exists an extensive error and stability analysis; in practical computations, however, one always uses a variable stepsize. It is well known that the varying stepsize affects the error and stability properties of the LMMs; the analysis of variable stepsize LMMs is generally much more complicated than for the fixed stepsize methods. Due to this, many of the computational algorithms in LMM codes are based on results from analyses for fixed stepsize LMMs or onestep methods, and the purpose pf these algorithms is partly to supply operating conditions that resemble as much as possible the properties of the fixed stepsize methods.



In this monograph we investigate different variable stepsize AMs and BDFs, with regard to method representation, solution of nonlinear equations, error estimation, and stepsize control. The methods are thoroughly derived and presented under a uniform taxonomy. In the error analysis the approach is to avoid premature Taylor approximations and crude norm bounds to be able to reveal some general properties of error propagation and error estimates for the different variable stepsize AMs and BDFs. The stepsize control is viewed from a control theoretical standpoint, where we, opposed to the conventional analysis, also take the errors' dependence on past stepsizes into account.



The results show that the assumptions, on which the conventional strategies rely, are not always fulfilled and, furthermore, that they can yield some undesirable secondary effects. We show that the predictors may have a severe impact on the behaviour of both method and error estimation properties. This will not only affect the choice of methods and predictors, but also several stages of the computational algorithms. (Less)
Please use this url to cite or link to this publication:
author
supervisor
opponent
  • Jackson, Ken, University of Toronto
organization
publishing date
type
Thesis
publication status
published
subject
keywords
linear multistep method, Initial value ODE, variable step method, Adams method, BDF, stepsize control, error analysis, Mathematics, Matematik
in
Doctoral Theses in Mathematical Sciences
volume
1999:6
pages
265 pages
publisher
Mathematics-Physics Section
defense location
N/A
defense date
1999-12-17 10:15:00
external identifiers
  • other:ISRN: LUNFNA-1001-1999
ISSN
1404-0034
ISBN
91-628-3898-9
language
English
LU publication?
yes
additional info
The information about affiliations in this record was updated in December 2015. The record was previously connected to the following departments: Numerical Analysis (011015004)
id
539a81a1-f8f0-47f1-b2cc-76590d2b1354 (old id 27702)
date added to LUP
2016-04-01 15:32:47
date last changed
2025-04-04 13:56:34
@phdthesis{539a81a1-f8f0-47f1-b2cc-76590d2b1354,
  abstract     = {{Linear multistep methods (LMMs) constitute a class of time-stepping methods for the solution of initial value ODEs; the most well-known methods of this class are the Adams methods (AMs) and the backward differentiation formulae (BDFs). For the fixed stepsize LMMs there exists an extensive error and stability analysis; in practical computations, however, one always uses a variable stepsize. It is well known that the varying stepsize affects the error and stability properties of the LMMs; the analysis of variable stepsize LMMs is generally much more complicated than for the fixed stepsize methods. Due to this, many of the computational algorithms in LMM codes are based on results from analyses for fixed stepsize LMMs or onestep methods, and the purpose pf these algorithms is partly to supply operating conditions that resemble as much as possible the properties of the fixed stepsize methods.<br/><br>
<br/><br>
In this monograph we investigate different variable stepsize AMs and BDFs, with regard to method representation, solution of nonlinear equations, error estimation, and stepsize control. The methods are thoroughly derived and presented under a uniform taxonomy. In the error analysis the approach is to avoid premature Taylor approximations and crude norm bounds to be able to reveal some general properties of error propagation and error estimates for the different variable stepsize AMs and BDFs. The stepsize control is viewed from a control theoretical standpoint, where we, opposed to the conventional analysis, also take the errors' dependence on past stepsizes into account.<br/><br>
<br/><br>
The results show that the assumptions, on which the conventional strategies rely, are not always fulfilled and, furthermore, that they can yield some undesirable secondary effects. We show that the predictors may have a severe impact on the behaviour of both method and error estimation properties. This will not only affect the choice of methods and predictors, but also several stages of the computational algorithms.}},
  author       = {{Sjö, Anders}},
  isbn         = {{91-628-3898-9}},
  issn         = {{1404-0034}},
  keywords     = {{linear multistep method; Initial value ODE; variable step method; Adams method; BDF; stepsize control; error analysis; Mathematics; Matematik}},
  language     = {{eng}},
  publisher    = {{Mathematics-Physics Section}},
  school       = {{Lund University}},
  series       = {{Doctoral Theses in Mathematical Sciences}},
  title        = {{Analysis of Computational Algorithms for Linear Multistep Methods}},
  volume       = {{1999:6}},
  year         = {{1999}},
}