{"has_accepted_license":"1","date_created":"2020-10-06T18:56:44Z","publication_status":"published","file_date_updated":"2020-10-07T14:01:58Z","author":[{"orcid":"https://orcid.org/0000-0002-5940-8057","last_name":"Offen","full_name":"Offen, Christian","id":"85279","first_name":"Christian"}],"main_file_link":[{"open_access":"1","url":"https://hdl.handle.net/10179/16155"}],"title":"Analysis of Hamiltonian boundary value problems and symplectic integration","extern":"1","abstract":[{"lang":"eng","text":"Ordinary differential equations (ODEs) and partial differential equations (PDEs) arise\r\nin most scientific disciplines that make use of mathematical techniques. As exact solutions are in general not computable, numerical methods are used to obtain approximate\r\nsolutions. In order to draw valid conclusions from numerical computations, it is crucial\r\nto understand which qualitative aspects numerical solutions have in common with the\r\nexact 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\r\nunder discretisation on long-term behaviour of motions is classically well known, in this\r\nthesis\r\n(a) the role of symplecticity for the bifurcation behaviour of solutions to Hamiltonian\r\nboundary value problems is explained. In parameter dependent systems at a bifurcation\r\npoint the solution set to a boundary value problem changes qualitatively. Bifurcation\r\nproblems are systematically translated into the framework of classical catastrophe theory. It is proved that existing classification results in catastrophe theory apply to\r\npersistent bifurcations of Hamiltonian boundary value problems. Further results for\r\nsymmetric settings are derived.\r\n(b) It is proved that to preserve generic bifurcations under discretisation it is necessary and sufficient to preserve the symplectic structure of the problem.\r\n(c) The catastrophe theory framework for Hamiltonian ODEs is extended to PDEs\r\nwith 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.\r\n(d) The potential of symplectic integration for infinite-dimensional Lie-Poisson systems (Burgers’ equation, KdV, fluid equations, . . . ) using Clebsch variables is analysed.\r\nIt is shown that the advantages of symplectic integration can outweigh the disadvantages of integrating over a larger phase space introduced by a Clebsch representation.\r\n(e) Finally, the preservation of variational structure of symmetric solutions in multisymplectic PDEs by multisymplectic integrators on the example of (phase-rotating)\r\ntravelling waves in the nonlinear wave equation is discussed."}],"supervisor":[{"first_name":"Robert I","full_name":"McLachlan, Robert I","last_name":"McLachlan"}],"place":"Palmerston North, New Zealand","file":[{"date_updated":"2020-10-07T14:01:58Z","file_name":"ths_all_signatures.pdf","file_id":"19948","access_level":"open_access","content_type":"application/pdf","date_created":"2020-10-06T18:54:53Z","creator":"coffen","relation":"main_file","file_size":19465740,"description":"A thesis presented in partial fulfilment of the requirements for the degree of Doctor of Philosophy in\nMathematics at Massey University, Manawatū, New Zealand.","title":"Thesis Christian Offen"}],"date_updated":"2022-01-06T06:54:16Z","alternative_title":["A thesis presented in partial fulfilment of the requirements for the degree of Doctor of Philosophy in Mathematics at Massey University, Manawatū, New Zealand."],"publisher":"Massey University","type":"dissertation","oa":"1","citation":{"short":"C. Offen, Analysis of Hamiltonian Boundary Value Problems and Symplectic Integration, Massey University, Palmerston North, New Zealand, 2020.","bibtex":"@book{Offen_2020, place={Palmerston North, New Zealand}, title={Analysis of Hamiltonian boundary value problems and symplectic integration}, publisher={Massey University}, author={Offen, Christian}, year={2020} }","ieee":"C. Offen, <i>Analysis of Hamiltonian boundary value problems and symplectic integration</i>. Palmerston North, New Zealand: Massey University, 2020.","ama":"Offen C. <i>Analysis of Hamiltonian Boundary Value Problems and Symplectic Integration</i>. Palmerston North, New Zealand: Massey University; 2020.","mla":"Offen, Christian. <i>Analysis of Hamiltonian Boundary Value Problems and Symplectic Integration</i>. Massey University, 2020.","chicago":"Offen, Christian. <i>Analysis of Hamiltonian Boundary Value Problems and Symplectic Integration</i>. Palmerston North, New Zealand: Massey University, 2020.","apa":"Offen, C. (2020). <i>Analysis of Hamiltonian boundary value problems and symplectic integration</i>. Palmerston North, New Zealand: Massey University."},"ddc":["510"],"_id":"19947","status":"public","year":"2020","user_id":"85279","language":[{"iso":"eng"}]}