Geometric integration concerns the analysis and construction of structure-preserving numerical methods for the long-time integration of differential equations that possess some geometric property, e.g. Hamiltonian or reversible systems. In choosing a structure-preserving method, it is important to consider its efficiency, stability, order, and ability to preserve qualitative properties of the differential system, such as time-reversal symmetry, symplecticity and energy-preservation. Commonly, the symmetric or symplectic Runge--Kutta methods, or the symmetric or G-symplectic linear multistep methods, are chosen as candidates for integration. In this thesis, a class of structure-preserving general linear methods (GLMs) is considered as an alternative choice. The research performed here includes the construction of a set of theoretical tools for analysing derivatives of B-series (a generalisation of Taylor series). These tools are then applied in the development of an a priori theory of parasitism for GLMs, which is used to prove bounds on the parasitic components of the method, and to derive algebraic conditions on the coefficients of the method that guarantee an extension of the time-interval of parasitism-free behaviour. A computational toolkit is also developed to help assist with this analysis, and for other analyses involving the manipulation of B-series and derivative B-series. High-order methods are constructed using a newly developed theory of composition for GLMs, which is an extension of the classical composition theory for one-step methods. A decomposition result for structure-preserving GLMs is also given which reveals that a memory-efficient implementation of these methods can be performed. This decomposition result is explored further, and it is shown that certain methods can be expressed as the composition of several LMMs. A variety of numerical experiments are performed on geometric differential systems to validate the theoretical results produced in this thesis, and to assess the competitiveness of these methods for long-time geometric integrations.
|Date of Award||29 Sep 2015|
|Supervisor||Adrian Hill (Supervisor)|