Christian Offen

Christian Offen staff profile picture

School of Fundamental Sciences
College of Sciences

Profile

Thesis Title
Geometric Integrators

Research Description
Hamiltonian ODEs and PDEs arise in classical mechanics as well as in fluid dynamics. Typically, Hamilton's equations of motion cannot be solved analytically such that reliable numerical methods are needed. The flow of Hamiltonian systems are symplectic. Symplecticty is a subtle geometric notion which is related to energy conservation of motions and other geometric properties as periodicity of motions and further topological properties of the phase portrait. To obtain qualitatively correct results it is desirable to preserve symplecticity when using a numerical method. While the significance of symplectic methods is well known and understood for initial value problems for Hamiltonian ODEs, I analyse the role of symplectic integration for solving boundary value problems and have shown that symplecticity is crucial when drawing bifurcation diagrams of solutions to Hamiltonian boundary value problems. Moreover, I analyse the effects of symplecticity in Hamiltonian PDEs. Having structure preserving algorithm available will enable researchers to obtain numerical results which are qualitatively correct despite calculating with limited accuracy.

Personal Description
With a strong background in Differential Geometry obtained during my Bachelor and Master studies at the University of Hamburg, Germany, and in numerical methods during an exchange semester at the University of Lund, Sweden, I joined the Applied Dynamics research group of Dist. Prof. Robert McLachlan in November 2016 to pursue my PhD. I love combining the world of pure and applied mathematics and applying techniques from Differential Geometry and Symplectic Geometry to questions arising in structure preserving numerical integration.

Supervisors
Distinguished Professor Robert McLachlan
Dr David Simpson

Publications

Published papers

(4) (with McLachlan, R., Tapely, B.) Symplectic integration of PDEs using Clebsch variables, Journal of Computational Dynamics, Volume 6, Number 1 (2019), https://dx.doi.org/10.3934/jcd.2019005
(3) (with McLachlan, R.) Symplectic integration of boundary value problems, Numerical Algorithms (2018), https://doi.org/10.1007/s11075-018-0599-7
(2) (with McLachlan, R.) Hamiltonian boundary value problems, conformal symplectic symmetries, and conjugate loci, Volume 48 (2018), 83-99, New Zealand Journal of Mathematics, https://arxiv.org/abs/1804.07479
(1) (with McLachlan, R.) Bifurcation of solutions to Hamiltonian boundary value problems, Nonlinearity, Volume 31, Number 6 (May 2018), https://doi.org/10.1088/1361-6544/aab630

Submitted papers

(7) (with Kreusser, L., McLachlan, R.) Detection of high codimensional bifurcations in variational PDEs, https://arxiv.org/abs/1903.02659
(6) Local intersections of Lagrangian manifolds correspond to catastrophe theory, https://arxiv.org/abs/1811.10165
(5) (with McLachlan, R.) Preservation of bifurcations of Hamiltonian boundary value problems under discretisation, https://arxiv.org/abs/1804.07468

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