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.

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.

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.

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.

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.