Offen, Christian
Christian
Offenhttps://orcid.org/0000-0002-5940-8057
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, ManawatÅ«, New Zealand.
Massey University
2020
2020-10-06T18:56:44Z
2021-03-16T14:57:49Z
dissertation
https://ris.uni-paderborn.de/record/19947
https://ris.uni-paderborn.de/record/19947.json
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Ordinary 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 (Burgersâ€™ 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.