Griffiths D.F., Higham D.J. Numerical Methods for Ordinary Differential Equations: Initial Value Problems

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Springer Undergraduate Mathematics Series

Advisory Board

M.A.J. Chaplain University of Dundee

K. Erdmann University of Oxford

A. MacIntyre Queen Mary, University of London

E. S

uli University of Oxford

For other titles published in this series, go to

www.springer.com/series/3423

M.R. Tehranchi University of Cambridge

J.F. Toland University of Bath

David F. Grifﬁths · Desmond J. Higham

Numerical Methods for

Ordinary Differential

Equations

Initial Value Problems

123

David F. Grifﬁths

Mathematics Division

University of Dundee

Dundee

dfg@maths.dundee.ac.uk

Desmond J. Higham

Department of Mathematics and Statistics

University of Strathclyde

Glasgow

djh@maths.strath.ac.uk

Springer Undergraduate Mathematics Series ISSN 1615-2085

ISBN 978-0-85729-147-9 e-ISBN 978-0-85729-148-6

DOI 10.1007/978-0-85729-148-6

Springer London Dordrecht Heidelberg New York

British Library Cataloguing in Publication Data

A catalogue record for this book is available from the British Library

Library of Congress Control Number: 2010937859

Mathematics Subject Classiﬁcation (2010): 65L05, 65L20, 65L06, 65L04, 65L07

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Preface

Diﬀerential equations, which describ e how quantities change across time or

space, arise naturally in science and engineering, and indeed in almost every

ﬁeld of study where measurements can be taken. For this reason, students

from a wide range of disciplines learn the fundamentals of calculus. They meet

diﬀerentiation and its evil twin, integration, and they are taught how to solve

some carefully chosen examples. These traditional pencil-and-paper techniques

provide an excellent means to put across the underlying theory, but they have

limited practical value. Most realistic mathematical models cannot be solved

in this way; instead, they must be dealt with by computational methods that

deliver approximate solutions.

Since the advent of digital computers in the mid 20th century, a vast amount

of eﬀort has been expended in designing, analysing and applying computational

techniques for diﬀerential equations. The topic has reached such a level of im-

portance that undergraduate students in mathematics, engineering, and physi-

cal sciences are typically exposed to one or more courses in the area. This book

provides material for the typical ﬁrst course—a short (20- to 30-hour) intro-

duction to numerical methods for initial-value ordinary diﬀerential equations

(ODEs). It is our belief that, in addition to e xposing students to core ideas in

numerical analysis, this type of course can also highlight the usefulness of tools

from calculus and analysis, in the best traditions of applied mathematics.

As a prerequisite, we assume a level of background knowledge consistent

with a standard ﬁrst course in calculus (Taylor series, chain rule, O(h) notation,

solving linear constant-coeﬃcient ODEs). Some key results are summarized in

Appendices B–D. For students wishing to brush up further on these topics,

there are many textb ooks on the market, including [12, 38, 64, 70], and a

plethora of on-line material can be reached via any reputable search engine.

viii Preface

There are already several undergraduate-level textbooks available that cover

numerical methods for initial-value ODEs, and many other general-purpos e nu-

merical analysis texts devote one or more chapters to this topic. However, we

feel that there is a niche for a well-focused and elementary text that concen-

trates on mathematical issues without losing sight of the applied nature of the

topic. This ﬁts in with the general philosophy of the Springer Undergraduate

Mathematics Series (SUMS), which aims to pro duce practical and concise texts

for undergraduates in mathematics and the sciences worldwide.

Based on many years of experience in teaching calculus and numerical analy-

sis to students in mathematics, engineering, and physical sciences, we have cho-

sen to follow the tried-and-tested format of Deﬁnition/Theorem/Proof, omit-

ting some of the more technical proofs. We believe that this type of structure

allows students to appreciate how a useful theory can be built up in a sequence

of logical steps. Within this formalization we have included a wealth of theo-

retical and computational examples that motivate and illustrate the ideas. The

material is broken down into manageable chapters, which are intended to repre-

sent one or two hours of lecturing. In keeping with the style of a typical lecture,

we are happy to repeat material (such as the speciﬁcation of our initial-value

ODE, or the general form of a linear multistep method) rather than frequently

cross-reference between separate chapters. Each chapter ends with a set of ex-

ercises that serve both to ﬁll in some details and allow students to test their

understanding. We have used a starring system: one star (?) for exercises with

short/simple answers, moving up to three stars (? ? ?) for longer/harder exer-

cises. Outline solutions to all exercises are available to authorized instructors

at the book’s website, which is available via http://www.springer.com. This

website also has links to useful resources and will host a list of corrections to

the book (fee l free to send us those).

To produce an elementary book like this, a number of tough decisions must

be made about what to leave out. Our omissions can be grouped into two

categories.

Theoretical We have glossed over the subject of existence and uniqueness of

solutions for ODEs. Rigorous analysis is beyond the scope of this book,

and very little understanding is imparted by simply stating without proof

the usual global Lipschitz conditions (which fail to be satisﬁed by most

realistic ODE models). Hence, our approach is always to assume that the

ODE has sm ooth solutions. From a numerical analysis perspective, we have

reluctantly shied away from a general-purpose Gronwall-style convergence

analysis and instead study global error propagation only for linear, scalar

constant-coeﬃcient ODEs. Convergence results are then state d, without

proof, more generally. This book has a strong emphasis on numerical stabil-

ity and qualitative properties of numerical methods hence; implicit methods

Preface ix

have a prominent role. Howeve r, we do not attempt to study general con-

ditions under which implicit recurrences have unique solutions, and how

they can be computed. Instead, we give simple examples and pose exer-

cises that illustrate some of the issues involved. Finally, although it would

be mathematically elegant to deal with systems of ODEs throughout the

bo ok, we have found that students are much more comfortable with s calar

problems. Hence, where possible, we do method development and analysis

on the scalar case, and then explain what, if anything, must be changed

to accommodate systems. To minimize confusion, we reserve a bold math-

ematical font (x, f , . . . ) for vector-valued quantities.

Practical This book does not set programming exercises: any computations

that are required can be done quickly with a calculator. We feel that this

is in keeping with the style of SUMS books; also, with the extensive range

of high-quality ODE software available in the public domain, it could be

argued that there is little need for students to write their own low-level

computer code. The main aim of this book is to give students an under-

standing of what goes on “under the hood” in scientiﬁc computing soft-

ware, and to equip them with a feel for the strengths and limitations of

numerical me thods. However, we strongly encourage readers to copy the

experiments in the book using whatever computational tools they have

available, and we hope that this material will encourage s tudents to take

further c ourses with a more practical scientiﬁc computing ﬂavour. We rec-

ommend the texts Ascher and Petzold [2] and Shampine et al. [62, 63] for

accessible treatments of the practical side of ODE computations and ref-

erences to state-of-the-art software. Most computations in this book were

carried out in the Matlab

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environment [34, Chapter 12].

By keeping the content tightly focused, we were able to make s pace for

some modern material that, in our opinion, deserves a higher proﬁle outside

the research literature. We have chosen four topics that (a) can be dealt with

using fairly simple mathematical concepts, and (b) give an indication of the

current open challenges in the area:

1. Nonlinear dynamics: spurious ﬁxed points and period two solutions.

2. Modiﬁed equations: construction, analysis, and interpretation.

3. Geometric integration: linear and quadratic invariants, symplecticness.

4. Stochastic diﬀerential equations: Brownian motion, Euler–Maruyama, weak

and strong convergence.

The ﬁeld of numerical methods for initial-value ODEs is fortunate to be

blessed with several high-quality, comprehensive research-level m onographs.