Table Of ContentXinyuan Wu
Bin Wang
Geometric Integrators
for Differential
Equations with Highly
Oscillatory Solutions
Geometric Integrators for Differential Equations
with Highly Oscillatory Solutions
Xinyuan Wu • Bin Wang
Geometric Integrators
for Differential Equations
with Highly Oscillatory
Solutions
XinyuanWu BinWang
DepartmentofMathematics SchoolofMathematicsandStatistics
NanjingUniversity Xi’anJiaotongUniversity
Nanjing,Jiangsu,China Xi’an,Shaanxi,China
ISBN978-981-16-0146-0 ISBN978-981-16-0147-7 (eBook)
https://doi.org/10.1007/978-981-16-0147-7
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Singapore
DedicatedtothememoryofProfessorFeng
Kangon thecentenaryof hisbirth.
TheprofoundandseminalcontributionsofProfessorFengKang,onsymplectic
geometricalgorithmsforHamiltoniansystems,haveopeneduparichandnewfield
ofnumericalmathematicsresearchinChinaandthroughouttheworld.
Foreword
The numerical integration of ordinary differential equations has a long and dis-
tinguished history inaugurated by Euler in the eighteenth century. The use of
numerical methods was of course extremely limited before computers became
availableanditisonlyaround1955thatthesubjecttookoff.Importanttheoretical
developments,whichusedverybeautifulmathematics,werecarriedoutbyLax(the
celebratedequivalencetheorem),Dahlquist(linearmultistepmethods)andButcher
(Runge–Kutta methods). The theory made it possible, starting in the 1960s, to
writepowerfulgeneral-purposesoftwarethatmayberoutinelyusedbypractitioners
to solve initial or boundary value problems. For the first time in history, it was
possibleforengineersandscientistsfacinga possiblyverycomplicated,nonlinear
differentialequationtofinditssolutionsinnexttonotimebypluggingitintoone
of the general-purposecodes that became widely available. This was a revolution
whoseimportanceiseasilyoverlookedbythosethathavegrownupsurroundedby
computersandsoftware.Oneofthestrengthsofthenumericaldifferentialequations
solversinsoftwarelibrariesisthat,asIhavementioned,theyaregeneralpurpose:
theyarecarefullycraftedblackboxesthatwilldealwithallinitialvalueproblems(to
bemoreprecise,oneneedsoneblackboxforstiffproblemsandasecondblackbox
for nonstiff problems). Without questioning the strengths and validity of general-
purpose numerical methods and software (that we may call “classical”), a very
different new paradigm appeared in the 1980s. The new paradigm brought many
innovativechanges.Intheclassicalapproachwhatwasdemandedofthenumerical
method was to find accurate numerical value of the solution at any desired time.
The new paradigm wanted to identify the long-time behaviour of the solutions
or perhaps the existence of conservation laws or some other qualitative feature
of the dynamics. The success of the classical software was based on a one-size-
fits-all approach; in the new paradigm, the numerical method and the software
were tailored to the applicationat hand, trying to capture as much as possible the
structureoftheproblembeingdealtwith.Inaddition,thenewparadigmoftenused
mathematicaltechniques, mainly from differentialgeometry,that had not hitherto
beenconnectedwithnumericalintegration.Finally,thenewparadigmintroducedor
popularizednewwaysofanalysingerrors,notablythemodifiedequationapproach.
vii
viii Foreword
In 1996, a State of the Art in Numerical Analysis conference was held in York
(England) with the purpose of analysing the main developments that had taken
placeinthe fieldofnumericalcomputationin thepreviousdecade.I wasaskedto
prepareasurveypresentationontheareaofordinarydifferentialequations,andfor
thattalk andthesubsequentpublication,I chosethetitle “GeometricIntegration”,
because I thought at the time that geometry was the unifying theme (or at least
oneof the unifyingthemes) in the new paradigm.The terminologycaughton and
it is now widely used in the literature. As discussed above, one of the salient
features of geometric integration is that specific classes of problems now take
centre stage. The first class to be considered was that of Hamiltonian problems,
whose studywas pioneeredbyProfessor Feng Kang;we commemoratedhis birth
centenary the year I wrote this foreword. Professor Feng Kang, who had made
other important contributions to mathematics, was the most important figure in
convincingtheinternationalcommunitythat,first,itisimportantfortheapplications
tocreatenumericalalgorithmsspecificallydesignedtosolveHamiltonianproblems
and that, second, geometric notion of symplecticness is essential to build the
new integrators—traditionalideas of stability and consistency were not sufficient.
Professor Feng Kang’s work was followed and is still being followed by many
hundredsorperhapsthousandsofpublicationsinChinaandthroughouttheworld.
Another area that has kept growing in importance within geometric integration is
thestudyofhighlyoscillatoryproblems:problemswherethesolutionsareperiodic
orquasiperiodicandhavetobestudiedintimeintervalsthatincludeanextremely
largenumberof periods. Examplesaboundin manifoldapplications:for instance,
we may wish to ascertain the future evolution of the solar system in intervals of
time where the planets have performedbillions of revolutions.The authorsof the
presentbook,ProfessorsXinyuanWuandBinWang,areamongthescientistswho,
in the internationalscene, have contributed most to the developmentand analysis
of geometric integrators for highly oscillatory differential equations, ordinary or
partial.Thisbookwillnodoubtbeavaluableadditiontoalonglistofpublications
thatstartedwiththeseminalpapersofProfessorFengKangandhisstudents.
President,RoyalAcademyofSciencesofSpain J.M.Sanz-Serna
ExcellenceChairinAppliedMathematics,Universidad
CarlosIIIdeMadrid
SIAMFellow
FellowoftheAmericanMathematicalSociety
FellowoftheInstituteofMathematicsanditsApplications
Preface
Differential equations that have highly oscillatory solutions arise in a variety of
fields in science and engineering such as astrophysics, classical and quantum
physics, and molecular dynamics, and their computation presents numerous chal-
lenges.Asisknown,theseequationscannotbesolvedefficientlyusingconventional
methods. Although notable progress has been made in numerical integrators for
highlyoscillatorydifferentialequations,itisnotobviousfortheseintegratorswhat
effectson their long-timebehaviourareproducedbypreservingcertain geometric
properties.Afurtherstudyofnovelgeometricintegratorshasbecomeincreasingly
importantinrecentyears.Theobjectiveofthisbookistoexplorefurthergeometric
integrators for highly oscillatory problems that can be formulated as systems of
ordinaryandpartialdifferentialequations.
This book has grown out of recent research work published in professional
journalsbythe researchgroupofthe authors.Thisbookis dividedintotwo parts.
Thefirstpartdealswithhighlyoscillatorysystemsofordinarydifferentialequations
(ODEs),andthesecondpartisconcernedwithtime-integrationofpartialdifferential
equations(PDEs)havingoscillatorysolutions.
The first part includes six chapters, dealing with highly oscillatory ODEs, and
the second part consists of eight chapters, providing some novel insights into
geometric integrators for PDEs. Chapter 1 is a review of oscillation-preserving
integrators for systems of second-order ODEs with highly oscillatory solutions.
As is known,continuous-stageRunge–Kutta–Nyström(RKN) methodshave been
developedfor this class of problems. Chapter 2 proposes and derives continuous-
stageextendedRunge–Kutta–Nyström(ERKN)integratorsforsecond-orderODEs
with highly oscillatory solutions. Since stability and convergence are essential
aspects of numerical analysis, we provide nonlinear stability and convergence
analysis of ERKN integrators for second-order ODEs with highly oscillatory
solutions in Chap.3. Poisson systems occur very frequently in physics, so in
Chap.4,weinvestigatefunctionallyfittedenergy-preservingintegratorsforPoisson
systems. We then consider exponential collocation methods for conservative or
dissipativesystemsinChap.5.Itisknownthatvariousdynamicalsystemsincluding
all Hamiltonian systems preserve volume in phase space, and hence we discuss
ix
x Preface
volume-preservingexponentialintegrators for first-order ODEs in Chap.6. Chap-
ter 7 analyses global error bounds of one-stage ERKN integrators for semilinear
wave equations. In Chap.8, we derive linearly fitted conservative (dissipative)
schemes for efficiently solving conservative (dissipative) nonlinear wave PDEs.
Chapter 9 focuses on the formulation and analysis of energy-preservingschemes
for solving high-dimensionalnonlinear Klein–Gordon equations. In Chap.10, we
introduce symmetric and arbitrarily high-order Hermite–Birkhoff time integrators
for solving nonlinearKlein–Gordonequations. Chapter 11 describesa symplectic
approximation with nonlinear stability and convergence analysis for efficiently
solving semilinear Klein–Gordon equations. Chapter 12 proposes and analyses
a continuous-stage modified leap-frog scheme for high-dimensional semilinear
Hamiltonian wave equations. Chapter 13 is concerned with semi-analytical expo-
nential RKN integrators for efficiently solving high-dimensional nonlinear wave
equationsbasedon fastFouriertransform(FFT) techniques.Chapter14considers
long-timemomentumandactionsbehaviourofenergy-preservingmethodsforwave
equations.
The presentation in this book provides some new perspectives of the sub-
ject which is based on theoretical derivations and mathematical analysis, facing
challenging scientific computational problems, and providing high-performance
numericalsimulations. In order to show the long-timenumericalbehaviourof the
simulation,alltheintegratorspresentedinthisbookhavebeentestedandverifiedon
highlyoscillatorysystemsfromawiderangeofapplicationsinthefieldofscience
andengineering.Theyaremoreefficientthanexistingschemesintheliteraturefor
differentialequationsthathavehighlyoscillatorysolutions.
We take this opportunity to thank all colleagues and friends for their selfless
helpduringthepreparationofthisbook.Amongthem,weparticularlyexpressour
heartfeltthankstoJohnButcheroftheUniversityofAuckland,ChristianLubichof
Universität Tübingen, Arieh Iserles of the University of Cambridge, J. M. Sanz-
Serna of Universidad Carlos III de Madrid, and Reinout Quispel of La Trobe
Universityfortheirencouragement.
The authors are also grateful to many colleagues and friends for reading the
manuscriptandfortheirvaluablecomments.SpecialthanksgotoRobertPengKong
ChanoftheUniversityofAuckland,QinShengofBaylorUniversity,JichunLiof
theUniversityofNevadaLasVegas,DavidMcLarenofLaTrobeUniversity,Adrian
HilloftheUniversityofBath,XiaowenChangofMcGillUniversity,JianlinXiaof
PurdueUniversityandMarcusDavidWebboftheUniversityofManchester.
Thanks go as well to the following colleagues and friends for their help and
supportinvariousforms:ZuheShen,JinxiZhao,YiqianWangandJianshengGeng
of Nanjing University; Fanwei Meng of Qufu Normal University; Yaolin Jiang
and Jing Gao of Xi’an Jiaotong University; Yongzhong Song, Yushun Wang and
QikuiDu of NanjingNormalUniversity;Chunwu Wang of NanjingUniversityof
AeronauticsandAstronautics;XinruWangofNanjingMedicalUniversity;Qiying
Wang of the University of Sydney; Shixiao Wang of the University of Auckland;
RobertMclachlanofMasseyUniversity;TianhaiTianofMonashUniversity;Choi-
Hong Lai of University of Greenwich; Jialin Hong, Zaijiu Shang, Yifa Tang and
Preface xi
YajuanSunoftheChineseAcademyofSciences;YuhaoCongofShanghaiCustoms
College;GuangdaHu ofShanghaiUniversity;ZhizhongSun andHongweiWu of
SoutheastUniversity;ShoufoLi,AiguoXiaoandLipingWenofXiangtanUniver-
sity;ChuanmiaoChenofHunanNormalUniversity;SiqingGanandXiaojieWang
ofCentralSouthUniversity;ChengjianZhang,ChengmingHuangandDongfangLi
ofHuazhongUniversityofScience&Technology;HongjiongTianandWansheng
Wang of Shanghai Normal University; Yongkui Zou of Jilin University; Jingjun
Zhaoof HarbinInstitute ofTechnology;XiaofeiZhaoand Jiwei Zhangof Wuhan
University;XiongYouofNanjingAgriculturalUniversity;WeiShiofNanjingTech
University, Qinghe Ming and Yonglei Fang of Zaozhuang University; Qinghong
LiofChuzhouUniversity,FanYang,XianyangZengandHongliYangofNanjing
InstituteofTechnology;KaiLiuofNanjingUniversityofFinanceandEconomics;
JiyongLiofHebeiNormalUniversity;andFazhanGengofChangshuInstituteof
Technology.
WewouldliketothankKaiHuandJiLuofortheirhelpwiththeediting,andthe
productionteamofSciencePressandSpringer-Verlag.
Wearegratefultoourfamilymembersfortheirloveandsupportthroughoutall
theseyears.
The work on this book was supported in part by the National Natural Science
FoundationofChinaunderGrantNo.11671200.
Nanjing,China XinyuanWu
Xi’an,China BinWang
