## Numerical methods for Ordinary Differential Equations

### Chapter 1 Differential and Difference Equations

This chapter aims to establish a common starting point for readers with a variety of backgrounds.  Many potential readers will have a much more sophisticated theoretical background than that assumed here.  Other possible readers will have a strong background in physical modelling but might not be so comfortable with the theory of differential equations.  There will also be readers who have a strong mathematical background but without much emphasis on differential equations.  It is hoped that working through this chapter will lead to an understanding of the basics of the subject together with an appreciation of some of the interesting and important differential equation systems which arise in physical and related problems.  Difference equations are not as fashionable as they once were in a standard mathematics curriculum but they are still important nonetheless.  The final sections of this chapter are intended to bring the reader to an introductory level in difference equations.

### Chapter 2 Numerical Differential Equation Methods

This chapter is a broad-ranging survey of standard methods for solving initial value problems.  It even goes beyond this and introduces the reader to some new methods.  However, this is a natural progression from the one-value, one-stage Euler method to multistage (Runge-Kutta) methods on one hand and multivalue (linear multistep) methods on the other.  Taylor series methods and hybrid (or general linear) methods simply take these generalizations a little further.  It is recommended to work carefully through this chapter before going on to the more advanced work in the later chapters.  This gives Chapter 2 a special role as a  preliminary course in its own right.

### Chapter 3 Runge-Kutta Methods

This is intended to be a comprehensive study of the theory and practice of Runge-Kutta methods.  The basic theory starts from the graph-theoretic approach; that is the Taylor expansions of the exact and approximate solutions are written in their natural form in terms of elementary differentials which in turn depend on rooted trees.  The connection between (rooted) trees and order conditions is exploited throughout the chapter and makes clear what would otherwise be arbitrary and unstructured.  The culmination of the tree approach is reached in Section 38 with the introduction of a group structure, which expands into an algebra.  This has applications outside the subject of this book but even here the applications are enough; it is argued that classical order is only a special case of the more general "effective order".  This theory also has applications in Chapter 5.

### Chapter 4 Linear Multistep Methods

Linear multistep methods are the great workhorses of numerical methods for differential equations and this chapter is a full treatment of them.  Much of the theory is the legacy of Germund Dahlquist and it is presented here in the slightly different style which the present author finds easy to work with.  Section 44 on order barriers uses not only the classical work of Dahlquist but also techniques inspired by the "order stars" of Hairer, Nørsett and Wanner.

### Chapter 5 General Linear Methods

The whole mish-mash of methods which combine the multistage nature of Runge-Kutta methods with the multivalue nature of linear multistep methods are referred to here as general linear methods.  There is now a mature and comprehensive theory of general linear methods, closely akin to the well-established theories of Runge-Kutta and linear multistep methods.  Of course purely formal generalizations of existing results and techniques are not as interesting as new insights and new practical algorithms.  The author believes that general linear methods are now marked by these achievements and the chapter presents some of this novel work.