Numerical methods for ordinary differential equations: initial value problems

274 Pages·2011·17.981 MB·English

by  Higham, Desmond J.;Griffiths, David Francis

Springer Undergraduate Mathematics Series
AdvisoryBoard
M.A.J.ChaplainUniversityofDundee
K.ErdmannUniversityofOxford
A.MacIntyreQueenMary,UniversityofLondon
E.Su¨liUniversityofOxford
M.R.TehranchiUniversityofCambridge
J.F.TolandUniversityofBath
Forothertitlespublishedinthisseries,goto
www.springer.com/series/3423
David F. Griffiths Desmond J. Higham
·
Numerical Methods for
Ordinary Differential
Equations
Initial Value Problems
123
DavidF.Griffiths DesmondJ.Higham
MathematicsDivision DepartmentofMathematicsandStatistics
UniversityofDundee UniversityofStrathclyde
Dundee Glasgow
[email protected] [email protected]
SpringerUndergraduateMathematicsSeriesISSN1615-2085
ISBN978-0-85729-147-9 e-ISBN978-0-85729-148-6
DOI10.1007/978-0-85729-148-6
SpringerLondonDordrechtHeidelbergNewYork
BritishLibraryCataloguinginPublicationData
AcataloguerecordforthisbookisavailablefromtheBritishLibrary
LibraryofCongressControlNumber:2010937859
MathematicsSubjectClassification(2010):65L05,65L20,65L06,65L04,65L07
c Springer-VerlagLondonLimited2010
⃝
Apart from any fair dealing for the purposes of research or private study, or criticism or review, as
permittedundertheCopyright,DesignsandPatentsAct1988,thispublicationmayonlybereproduced,
stored or transmitted, in any form or by any means, with the prior permission in writing of the
publishers,orinthecaseofreprographicreproductioninaccordancewiththetermsoflicensesissued
bytheCopyrightLicensingAgency.Enquiriesconcerningreproductionoutsidethosetermsshouldbe
senttothepublishers.
Theuseofregisterednames,trademarks,etc.,inthispublicationdoesnotimply,evenintheabsenceofa
specificstatement,thatsuchnamesareexemptfromtherelevantlawsandregulationsandthereforefree
forgeneraluse.
Thepublishermakesnorepresentation,expressorimplied,withregardtotheaccuracyoftheinformation
containedinthisbookandcannotacceptanylegalresponsibilityorliabilityforanyerrorsoromissions
thatmaybemade.
Printedonacid-freepaper
SpringerispartofSpringerScience+BusinessMedia(www.springer.com)
To the Dundee Numerical Analysis Group past and present,
of which we are proud to have been a part.
Preface
Di↵erential equations, which describe 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
di↵erentiation and its evil twin, integration, and they are taught how to solve
somecarefullychosenexamples.Thesetraditionalpencil-and-papertechniques
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.
Sincetheadventofdigitalcomputersinthemid20thcentury,avastamount
ofe↵orthasbeenexpendedindesigning,analysingandapplyingcomputational
techniques for di↵erential equations. The topic has reached such a level of im-
portancethatundergraduatestudentsinmathematics,engineering,andphysi-
calsciencesaretypicallyexposedtooneormorecoursesinthearea.Thisbook
provides material for the typical first course—a short (20- to 30-hour) intro-
duction to numerical methods for initial-value ordinary di↵erential equations
(ODEs). It is our belief that, in addition to exposing students to core ideas in
numericalanalysis,thistypeofcoursecanalsohighlighttheusefulness oftools
from calculus and analysis, in the best traditions of applied mathematics.
As a prerequisite, we assume a level of background knowledge consistent
withastandardfirstcourseincalculus(Taylorseries,chainrule, (h)notation,
O
solving linear constant-coe�cient ODEs). Some key results are summarized in
Appendices B–D. For students wishing to brush up further on these topics,
there are many textbooks 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
Therearealreadyseveralundergraduate-leveltextbooksavailablethatcover
numericalmethodsforinitial-valueODEs,andmanyothergeneral-purposenu-
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
MathematicsSeries(SUMS),whichaimstoproducepractical andconcise texts
for undergraduates in mathematics and the sciences worldwide.
Basedonmanyyearsofexperienceinteachingcalculusandnumericalanaly-
sistostudentsinmathematics,engineering,andphysicalsciences,wehavecho-
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
allowsstudentstoappreciatehowausefultheorycanbebuiltupinasequence
of logical steps. Within this formalization we have included a wealth of theo-
reticalandcomputationalexamplesthatmotivateandillustratetheideas.The
materialisbrokendownintomanageablechapters,whichareintendedtorepre-
sentoneortwohoursoflecturing.Inkeepingwiththestyleofatypicallecture,
we are happy to repeat material (such as the specification of our initial-value
ODE,or the generalform ofa linearmultistepmethod) ratherthanfrequently
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 (feel free to send us those).
Toproduceanelementarybooklikethis,anumberoftoughdecisionsmust
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
ODEhassmoothsolutions.Fromanumericalanalysisperspective,wehave
reluctantly shied away from a general-purpose Gronwall-style convergence
analysis and instead study global error propagation only for linear, scalar
constant-coe�cient ODEs. Convergence results are then stated, without
proof,moregenerally.Thisbookhasastrongemphasisonnumericalstabil-
ityandqualitativepropertiesofnumericalmethodshence;implicitmethods
Preface ix
have a prominent role. However, 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
book, we have found that students are much more comfortable with scalar
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 methods. 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 students to take
further courses 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 Matlabc environment [34, Chapter 12].
�
By keeping the content tightly focused, we were able to make space 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. Stochasticdi↵erentialequations:Brownianmotion,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 monographs.
x Preface
Ratherthanpepperthisbookwithreferencestothesameclassictexts,itseems
more appropriate to state here that, for all general numerical ODE questions,
the oracles are [6], [28] and [29]: that is,
J. C. Butcher, Numerical Methods for Ordinary Di↵erential Equations, 2nd
edition, Wiley, 2008,
E. Hairer, S. P. Nørsett and G. Wanner, Solving Ordinary Di↵erential Equa-
tions I: Nonsti↵ Problems, 2nd edition, Springer, 1993,
E. Hairer and G. Wanner, Solving Ordinary Di↵erential Equations II: Sti↵
and Di↵erential-Algebraic Problems, 2nd edition, Springer, 1996.
To learn more about the topics touched on in Chapters 12–16, we recom-
mend Stuart and Humphries [65] for numerical dynamics, Hairer et al. [26],
Leimkuhler and Reich [45], and Sanz-Serna and Calvo [61] for geometric in-
tegration, and Kloeden and Platen [42] and Milstein and Tretyakov [52] for
stochastic di↵erential equations.
Acknowledgements We thank Niall Dodds, Christian Lubich and J. M.
(Chus) Sanz-Serna for their careful reading of the manuscript. We are also
grateful to the anonymous reviewers used by the publisher, who made many
valuable suggestions.
We are particularly indebted to our former colleagues A. R. (Ron) Mitchell
and J. D. (Jack) Lambert for their influence on us in all aspects of numerical
di↵erential equations.
Itwouldalsoberemissofusnottomentionthepatienceandhelpfulnessof
our editors, Karen Borthwick and Lauren Stoney, who did much to encourage
us towards the finishing line.
Finally, we thank members of our families, Anne, Sarah, Freya, Philip,
Louise,Oliver,Catherine,Theo,Sophie,andLucas,fortheirloveandsupport.
DFG, DJH
August 2010
Contents
1. ODEs—An Introduction.................................... 1
1.1 Systems of ODEs......................................... 5
1.2 Higher Order Di↵erential Equations......................... 12
1.3 Some Model Problems .................................... 14
2. Euler’s Method............................................. 19
2.1 A Preliminary Example ................................... 20
2.1.1 Analysing the Numbers ............................. 21
2.2 Landau Notation ......................................... 22
2.3 The General Case ........................................ 23
2.4 Analysing the Method .................................... 25
2.5 Application to Systems.................................... 29
3. The Taylor Series Method .................................. 33
3.1 Introduction ............................................. 33
3.2 An Order-Two Method: TS(2) ............................. 34
3.2.1 Commentary on the Construction .................... 36
3.3 An Order-p Method: TS(p) ................................ 36
3.4 Convergence ............................................. 37
3.5 Application to Systems.................................... 38
3.6 Postscript ............................................... 39
4. Linear Multistep Methods—I............................... 43
4.1 Introduction ............................................. 43
4.1.1 The Trapezoidal Rule............................... 44
4.1.2 The 2-step Adams–Bashforth method: AB(2) .......... 45