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.
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.
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.
Original language  English 

Qualification  Doctor 
Awarding Institution 

Supervisors/Advisors 

Award date  1999 Dec 17 
Publisher  
ISBN (Print)  9162838989 
Publication status  Published  1999 
Bibliographical note
Defence detailsDate: 19991217
Time: 10:15
Place: N/A
External reviewer(s)
Name: Jackson, Ken
Title: [unknown]
Affiliation: University of Toronto

The information about affiliations in this record was updated in December 2015.
The record was previously connected to the following departments: Numerical Analysis (011015004)
Subject classification (UKÄ)
 Mathematics
Free keywords
 linear multistep method
 Initial value ODE
 variable step method
 Adams method
 BDF
 stepsize control
 error analysis
 Mathematics
 Matematik