Long-term behaviour of G-symplectic methods

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dc.contributor.advisor Butcher, J en
dc.contributor.advisor Chan, R en
dc.contributor.author Habib, Yousaf en
dc.date.accessioned 2011-03-30T21:59:05Z en
dc.date.issued 2010 en
dc.identifier.uri http://hdl.handle.net/2292/6641 en
dc.description.abstract There has been a recent revival of interest in structure preserving numerical methods for ordinary differential equations having quadratic invariants. Much work has been done for Runge-Kutta and multistep methods and there exist excellent symplectic integrators among Runge-Kutta methods. General linear methods provide a unifying framework for these traditional methods but, because of their multivalue nature we cannot hope for true conservation of quadratic invariants. However, not everything is lost and we can still search for G-symplectic general linear methods taking account of the underlying invariants. The multivalue nature of general linear methods exposes them to parasitic solutions. The corruption of the numerical solution is partly due to the parasitic growth parameter and partly due to the differential equation system being susceptible to parasitism. Two control strategies have been employed to contain this situation. One, where the effective parasitic growth parameter of a composition of different G-symplectic methods is forced to remain bounded. Several possible composition techniques can be used of which one is employed in this thesis and further reference is provided in the conclusions. The other strategy is to construct methods where parasitic growth parameter is zero by design. The construction of a method with four stages and three output values and a search for a suitable starting method with algebraic analysis using rooted trees constitute an important aspect of this thesis. These strategies are investigated using various implementations for Hamiltonian and structure preserving systems and compared with a traditional symplectic method. This provides encouraging results for the G-symplectic general linear methods. The new methods provide an alternative to the well established symplectic one step methods. The foundation for the search of such methods is laid out in this thesis and it is anticipated that these methods can be implemented for serious real world problems with confidence. en
dc.publisher ResearchSpace@Auckland en
dc.relation.ispartof PhD Thesis - University of Auckland en
dc.relation.isreferencedby UoA99208971514002091 en
dc.rights Items in ResearchSpace are protected by copyright, with all rights reserved, unless otherwise indicated. Previously published items are made available in accordance with the copyright policy of the publisher. en
dc.rights.uri https://researchspace.auckland.ac.nz/docs/uoa-docs/rights.htm en
dc.rights.uri http://creativecommons.org/licenses/by-nc-sa/3.0/nz/ en
dc.title Long-term behaviour of G-symplectic methods en
dc.type Thesis en
thesis.degree.discipline Applied Mathematics en
thesis.degree.grantor The University of Auckland en
thesis.degree.level Doctoral en
thesis.degree.name PhD en
dc.rights.holder Copyright: The author en
pubs.peer-review false en
pubs.elements-id 208375 en
pubs.record-created-at-source-date 2011-03-31 en

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