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Title: Non-standard discretizations of differential equations
Author: Towler, Kim
ISNI:       0000 0004 7227 7615
Awarding Body: University of Kent
Current Institution: University of Kent
Date of Award: 2015
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This thesis explores non-standard numerical integration methods for a range of non-linear systems of differential equations with a particular interest in looking for the preservation of various features when moving from the continuous system to a discrete setting. Firstly the exsiting non-standard schemes such as one discovered by Hirota and Kimura (and also Kahan) [21, 32] will be presented along with general rules for creating an effective numerical integration scheme devised by Mickens [40]. We then move on to the specific example of the Lotka-Volterra system and present a method for finding the most general forms of a non-standard scheme that is both symplectic and birational. The resulting three schemes found through this method have also been discovered through an alternative method by Roeger in [52]. Next we look at discretizing examples of 3-dimensional bi-Hamiltonian systems from a list given by G¨umral and Nutku [18] using the Hirota-Kimura/Kahan method followed by the same method applied to the H´enon-Heiles case (ii) system. The B¨acklund transformation for the H´enon-Heiles is also considered. Finally chapter 6 looks at systems with cubic vector fields and limit cycles with an aim to find the most general form of a non-standard scheme for two examples. First we look at a trimolecular system and then a Hamiltonian system that has a quartic potential.
Supervisor: Hone, Andrew Sponsor: Not available
Qualification Name: Thesis (Ph.D.) Qualification Level: Doctoral
EThOS ID:  DOI: Not available