Numerical
methods for Ordinary Differential Equations
Contents
according to John Butcher
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.
10 Differential Equation Problems
11 Differential Equation Theory
12 Further Evolutionary Problems
13 Difference
Equation Problem
14 Difference
Equation Theory
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.
20
The Euler Method
21 Analysis
of the Euler Method
22 Generalizations
of the Euler Method
23 Runge-Kutta
Methods
24 Linear
Multistep Methods
25 Taylor
Series Methods
26 Hybrid
Methods
27 Introduction
to implementation
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.
30 Preliminaries
31 Order Conditions
32 Low Order Explicit Methods
33 Runge-Kutta
Methods with Error Estimates
34 Implicit
Runge-Kutta Methods
35 Stability
of Implicit Runge-Kutta Methods
36 Implementable
Implicit Runge-Kutta Methods
37 Symplectic
Runge-Kutta Methods
38 Algebraic
Properties of Runge-Kutta Methods
39 Implementation
Issues
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.
40 Preliminaries
41 The Order of Linear Multistep Methods
42 Errors and Error Growth
43 Stability
Characteristics
44 Order
and Stability Barriers
45 One-Leg
Methods and G-stability
46 Implementation
Issues
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.
50 Representing Methods in General Linear Form
51 Consistency, Stability and Convergence
52 The Stability of General Linear Methods
53 The
Order of General Linear Methods
54 Methods
with Runge-Kutta stability
55 Methods
with Inherent Runge--Kutta Stability