Geometric Integrators for Differential Equations with Highly Oscillatory Solutions PDF
Preview Geometric Integrators for Differential Equations with Highly Oscillatory Solutions
Xinyuan 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 JointlypublishedwithSciencePress TheprinteditionisnotforsaleinChina(Mainland).CustomersfromChina(Mainland)pleaseorderthe printbookfrom:SciencePress. ©TheEditor(s)(ifapplicable)andTheAuthor(s),underexclusivelicensetoSpringerNatureSingapore PteLtd.2021 Thisworkissubjecttocopyright.AllrightsaresolelyandexclusivelylicensedbythePublisher,whether thewhole orpart ofthematerial isconcerned, specifically therights oftranslation, reprinting, reuse ofillustrations, recitation, broadcasting, reproductiononmicrofilmsorinanyotherphysicalway,and transmissionorinformationstorageandretrieval,electronicadaptation,computersoftware,orbysimilar ordissimilarmethodologynowknownorhereafterdeveloped. Theuseofgeneraldescriptivenames,registerednames,trademarks,servicemarks,etc.inthispublication doesnotimply,evenintheabsenceofaspecificstatement,thatsuchnamesareexemptfromtherelevant protectivelawsandregulationsandthereforefreeforgeneraluse. Thepublisher,theauthors,andtheeditorsaresafetoassumethattheadviceandinformationinthisbook arebelievedtobetrueandaccurateatthedateofpublication.Neitherthepublishernortheauthorsor theeditorsgiveawarranty,expressedorimplied,withrespecttothematerialcontainedhereinorforany errorsoromissionsthatmayhavebeenmade.Thepublisherremainsneutralwithregardtojurisdictional claimsinpublishedmapsandinstitutionalaffiliations. ThisSpringerimprintispublishedbytheregisteredcompanySpringerNatureSingaporePteLtd. The registered company address is: 152 Beach Road, #21-01/04 Gateway East, Singapore 189721, 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
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