Analysis of Computational Algorithms for Linear Multistep Methods
(1999) In Doctoral Thesis in Mathematical Sciences 1999:6. Abstract
 Linear multistep methods (LMMs) constitute a class of timestepping methods for the solution of initial value ODEs; the most wellknown 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 timestepping methods for the solution of initial value ODEs; the most wellknown 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:
http://lup.lub.lu.se/record/27702
 author
 Sjö, Anders ^{LU}
 opponent

 Jackson, Ken, University of Toronto
 organization
 publishing date
 1999
 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 Thesis in Mathematical Sciences
 volume
 1999:6
 pages
 265 pages
 publisher
 MathematicsPhysics Section
 defense location
 N/A
 defense date
 19991217 10:15
 external identifiers

 other:ISRN: LUNFNA10011999
 ISSN
 14040034
 ISBN
 9162838989
 language
 English
 LU publication?
 yes
 id
 539a81a1f8f047f1b2cc76590d2b1354 (old id 27702)
 date added to LUP
 20070607 15:41:15
 date last changed
 20160919 08:44:52
@phdthesis{539a81a1f8f047f1b2cc76590d2b1354, abstract = {Linear multistep methods (LMMs) constitute a class of timestepping methods for the solution of initial value ODEs; the most wellknown 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 = {9162838989}, issn = {14040034}, keyword = {linear multistep method,Initial value ODE,variable step method,Adams method,BDF,stepsize control,error analysis,Mathematics,Matematik}, language = {eng}, pages = {265}, publisher = {MathematicsPhysics Section}, school = {Lund University}, series = {Doctoral Thesis in Mathematical Sciences}, title = {Analysis of Computational Algorithms for Linear Multistep Methods}, volume = {1999:6}, year = {1999}, }