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Introduction
Our main interest in this thesis is the study of numerical methods for dynamical systems de¯ned by (ordinary) di®erential equations. Prob- lems as diverse as the simulation of planetary interactions, °uid °ows [10] and mechanics [43], chemical reactions [16],[40], biological pattern formulation [2], [18], [33] and economic markets can all be modelled as dynamical systems [41]. For further applications of dynamical systems see [44]. In most of the systems modelled, all rates of change are as- sumed to be time independent, which makes the corresponding system autonomous.
Dynamical systems are concerned primarily with making qualita- tive study about the behaviour of systems which evolve in time given knowledge about the initial state of the system itself. It is important to know and study essential qualitative properties of the systems or more precisely their dynamics. Such properties include among others:
the type of ¯xed points, oscillatory solutions, monotonicity of solutions, conservation of energy, dissipativity or dispersion of solution, positivity and boundedness of solutions. Our standard reference for dynamical systems is Stuart and Humphries [41] while Lambert [22] will also be used for numerical methods for ordinary di®erential equations. The framework of the study will include a wide range of concrete linear and non-linear models such as: logistic equation, decay equation, Hamil- tonian system in ordinary di®erential equations as well as the Fisher equation, the reaction-di®usion equation in partial di®erential equa- tions.
Existence theory is extensively developed for di®erential equations. However, most di®erential equations have no analytical solutions. As a result numerical methods are of fundamental importance in gaining understanding of dynamical systems. For contemporary numerical ana- lysts, the understanding of di®erential equations from numerical meth- ods is often limited to the study of their consistency, (zero-) stability and convergence. Unfortunately such classical numerical methods do not guarantee that the dynamics of the systems are replicated. This explains why we use the monograph [41] as our standard reference on dynamical systems, since it is one of a few classical books emphasizing the similar properties of the exact solutions that numerical schemes exhibit.
1 Introduction
2 Dynamical Systems
2.1 Introduction
2.2 Continuous Dynamical Systems
2.2.1 Generalities
2.2.2 Qualitative Properties
2.3 Discrete Dynamical Systems
2.3.1 Generalities
2.3.2 Qualitative Properties
3 Finite Di®erence Methods
3.1 Introduction
3.2 Basic Concepts
3.3 Linear Multi-step Methods
3.4 Runge-Kutta Methods
3.5 Absolute Stability
3.5.1 Linear Multi-step Methods
3.5.2 Runge-Kutta Methods
3.6 Numerical Methods as Dynamical Systems
3.7 Theta Methods
4 Non-standard Finite Di®erence Methods
4.1 Introduction
4.2 Generalities
4.3 Elementary Stable Schemes
4.4 Dissipative Non-standard Theta Methods
4.5 Energy Preserving Discrete Schemes
5 Non-standard Finite Di®erence Schemes for Reaction-
Di®usion Equations
6 Conclusion
Bibliography
Summary
