Collocation methods for a class of second order initial value problems with oscillatory solutions
We derive and analyse two families of multistep collocation methods for periodic initial-value problems of the form y" = f(x, y); y((^x)o) = yo, y(^1)(xo) = zo involving ordinary differential equations of second order in which the first derivative does not appear explicitly. A survey of recent...
Main Author: | |
---|---|
Published: |
Durham University
1993
|
Subjects: | |
Online Access: | http://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.359136 |
id |
ndltd-bl.uk-oai-ethos.bl.uk-359136 |
---|---|
record_format |
oai_dc |
spelling |
ndltd-bl.uk-oai-ethos.bl.uk-3591362015-03-19T05:36:37ZCollocation methods for a class of second order initial value problems with oscillatory solutionsBooth, Andrew S.1993We derive and analyse two families of multistep collocation methods for periodic initial-value problems of the form y" = f(x, y); y((^x)o) = yo, y(^1)(xo) = zo involving ordinary differential equations of second order in which the first derivative does not appear explicitly. A survey of recent results and proposed numerical methods is given in chapter 2. Chapter 3 is devoted to the analysis of a family of implicit Chebyshev methods proposed by Panovsky k Richardson. We show that for each non-negative integer r, there are two methods of order 2r from this family which possess non-vanishing intervals of periodicity. The equivalence of these methods with one-step collocation methods is also established, and these methods are shown to be neither P-stable nor symplectic. In chapters 4 and 5, two families of multistep collocation methods are derived, and their order and stability properties are investigated. A detailed analysis of the two-step symmetric methods from each class is also given. The multistep Runge-Kutta-Nystrom methods of chapter 4 are found to be difficult to analyse, and the specific examples considered are found to perform poorly in the areas of both accuracy and stability. By contrast, the two-step symmetric hybrid methods of chapter 5 are shown to have excellent stability properties, in particular we show that all two-step 27V-point methods of this type possess non-vanishing intervals of periodicity, and we give conditions under which these methods are almost P-stable. P-stable and efficient methods from this family are obtained and demonstrated in numerical experiments. A simple, cheap and effective error estimator for these methods is also given.519Runge-Kutta-Nystroem methodDurham Universityhttp://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.359136http://etheses.dur.ac.uk/5664/Electronic Thesis or Dissertation |
collection |
NDLTD |
sources |
NDLTD |
topic |
519 Runge-Kutta-Nystroem method |
spellingShingle |
519 Runge-Kutta-Nystroem method Booth, Andrew S. Collocation methods for a class of second order initial value problems with oscillatory solutions |
description |
We derive and analyse two families of multistep collocation methods for periodic initial-value problems of the form y" = f(x, y); y((^x)o) = yo, y(^1)(xo) = zo involving ordinary differential equations of second order in which the first derivative does not appear explicitly. A survey of recent results and proposed numerical methods is given in chapter 2. Chapter 3 is devoted to the analysis of a family of implicit Chebyshev methods proposed by Panovsky k Richardson. We show that for each non-negative integer r, there are two methods of order 2r from this family which possess non-vanishing intervals of periodicity. The equivalence of these methods with one-step collocation methods is also established, and these methods are shown to be neither P-stable nor symplectic. In chapters 4 and 5, two families of multistep collocation methods are derived, and their order and stability properties are investigated. A detailed analysis of the two-step symmetric methods from each class is also given. The multistep Runge-Kutta-Nystrom methods of chapter 4 are found to be difficult to analyse, and the specific examples considered are found to perform poorly in the areas of both accuracy and stability. By contrast, the two-step symmetric hybrid methods of chapter 5 are shown to have excellent stability properties, in particular we show that all two-step 27V-point methods of this type possess non-vanishing intervals of periodicity, and we give conditions under which these methods are almost P-stable. P-stable and efficient methods from this family are obtained and demonstrated in numerical experiments. A simple, cheap and effective error estimator for these methods is also given. |
author |
Booth, Andrew S. |
author_facet |
Booth, Andrew S. |
author_sort |
Booth, Andrew S. |
title |
Collocation methods for a class of second order initial value problems with oscillatory solutions |
title_short |
Collocation methods for a class of second order initial value problems with oscillatory solutions |
title_full |
Collocation methods for a class of second order initial value problems with oscillatory solutions |
title_fullStr |
Collocation methods for a class of second order initial value problems with oscillatory solutions |
title_full_unstemmed |
Collocation methods for a class of second order initial value problems with oscillatory solutions |
title_sort |
collocation methods for a class of second order initial value problems with oscillatory solutions |
publisher |
Durham University |
publishDate |
1993 |
url |
http://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.359136 |
work_keys_str_mv |
AT boothandrews collocationmethodsforaclassofsecondorderinitialvalueproblemswithoscillatorysolutions |
_version_ |
1716741972731887616 |