NON-STANDARD DISCRETIZATIONS OF DIFFERENTIAL EQUATIONS

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.