Analysis of Hamiltonian boundary value problems and symplectic integration: a thesis presented in partial fulfilment of the requirements for the degree of Doctor of Philosophy in Mathematics at Massey University, Manawatu, New Zealand

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dc.confidentialEmbargo : Noen_US
dc.contributor.advisorMcLachlan, Robert
dc.contributor.authorOffen, Christian
dc.date.accessioned2020-07-23T22:11:03Z
dc.date.accessioned2021-03-15T20:44:02Z
dc.date.available2020-07-23T22:11:03Z
dc.date.available2021-03-15T20:44:02Z
dc.date.issued2020
dc.descriptionListed in 2020 Dean's List of Exceptional Thesesen
dc.descriptionCopyright is owned by the Author of the thesis. Permission is given for a copy to be downloaded by an individual for research and private study only. The thesis may not be reproduced elsewhere without the permission of the Author.en
dc.description.abstractOrdinary differential equations (ODEs) and partial differential equations (PDEs) arise in most scientific disciplines that make use of mathematical techniques. As exact solutions are in general not computable, numerical methods are used to obtain approximate solutions. In order to draw valid conclusions from numerical computations, it is crucial to understand which qualitative aspects numerical solutions have in common with the exact solution. Symplecticity is a subtle notion that is related to a rich family of geometric properties of Hamiltonian systems. While the effects of preserving symplecticity under discretisation on long-term behaviour of motions is classically well known, in this thesis (a) the role of symplecticity for the bifurcation behaviour of solutions to Hamiltonian boundary value problems is explained. In parameter dependent systems at a bifurcation point the solution set to a boundary value problem changes qualitatively. Bifurcation problems are systematically translated into the framework of classical catastrophe theory. It is proved that existing classification results in catastrophe theory apply to persistent bifurcations of Hamiltonian boundary value problems. Further results for symmetric settings are derived. (b) It is proved that to preserve generic bifurcations under discretisation it is necessary and sufficient to preserve the symplectic structure of the problem. (c) The catastrophe theory framework for Hamiltonian ODEs is extended to PDEs with variational structure. Recognition equations for $A$-series singularities for functionals on Banach spaces are derived and used in a numerical example to locate high-codimensional bifurcations. (d) The potential of symplectic integration for infinite-dimensional Lie-Poisson systems (Burgers' equation, KdV, fluid equations,...) using Clebsch variables is analysed. It is shown that the advantages of symplectic integration can outweigh the disadvantages of integrating over a larger phase space introduced by a Clebsch representation. (e) Finally, the preservation of variational structure of symmetric solutions in multisymplectic PDEs by multisymplectic integrators on the example of (phase-rotating) travelling waves in the nonlinear wave equation is discussed.en_US
dc.identifier.urihttp://hdl.handle.net/10179/16155
dc.publisherMassey Universityen_US
dc.rightsThe Authoren_US
dc.subjectBoundary value problemsen
dc.subjectHamiltonian systemsen
dc.subjectSymplectic geometryen
dc.subjectDean's List of Exceptional Thesesen
dc.subject.anzsrc490303 Numerical solution of differential and integral equationsen
dc.titleAnalysis of Hamiltonian boundary value problems and symplectic integration: a thesis presented in partial fulfilment of the requirements for the degree of Doctor of Philosophy in Mathematics at Massey University, Manawatu, New Zealanden_US
dc.typeThesisen_US
massey.contributor.authorOffen, Christianen_US
thesis.degree.disciplineMathematicsen_US
thesis.degree.grantorMassey Universityen_US
thesis.degree.levelDoctoralen_US
thesis.degree.nameDoctor of Philosophy (PhD)en_US
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