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Structure-Preserving Algorithms for Oscillatory Differential Equations Xinyuan Wu (cid:2) Xiong You (cid:2) Bin Wang Structure-Preserving Algorithms for Oscillatory Differential Equations XinyuanWu XiongYou NanjingUniversity NanjingAgriculturalUniversity Nanjing,China Nanjing,China BinWang NanjingUniversity Nanjing,China ISBN978-3-642-35337-6 ISBN978-3-642-35338-3(eBook) DOI10.1007/978-3-642-35338-3 SpringerHeidelbergNewYorkDordrechtLondon JointlypublishedwithSciencePressBeijing ISBN:978-7-03-035520-1SciencePressBeijing LibraryofCongressControlNumber:2013931848 ©SciencePressBeijingandSpringer-VerlagBerlinHeidelberg2013 Thisworkissubjecttocopyright.AllrightsarereservedbythePublisher,whetherthewholeorpartof thematerialisconcerned,specificallytherightsoftranslation,reprinting,reuseofillustrations,recitation, broadcasting,reproductiononmicrofilmsorinanyotherphysicalway,andtransmissionorinformation storageandretrieval,electronicadaptation,computersoftware,orbysimilarordissimilarmethodology nowknownorhereafterdeveloped.Exemptedfromthislegalreservationarebriefexcerptsinconnection with reviews or scholarly analysis or material supplied specifically for the purpose of being entered and executed on a computer system, for exclusive use by the purchaser of the work. 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Printedonacid-freepaper SpringerispartofSpringerScience+BusinessMedia(www.springer.com) Preface Effectivenumericalsolutionofdifferentialequations,althoughasoldasdifferential equations themselves, has been a great challenge to numerical analysts, scientists and engineers for centuries. In recent decades, it has been universally acknowl- edgedthatdifferentialequationsarisinginscienceandengineeringoftenhavecer- tainstructuresthatrequirepreservationbythenumericalintegrators.Beginningwith thesymplecticintegrationofR.deVogelaere(1956),R.D.Ruth(1983),FengKang (1985), J.M. Sanz-Serna (1988), E. Hairer (1994) and others, structure-preserving computation, or geometric numerical integration, has become one of the central fieldsofnumericaldifferentialequations.Geometricnumericalintegrationaimsat thepreservationofthephysicalorgeometricfeaturesoftheexactflowofthesystem inlong-termcomputation,suchasthesymplecticstructureofHamiltoniansystems, energyandmomentumofdynamicalsystems,time-reversibilityofconservativeme- chanicalsystems,oscillatoryandhighoscillatorysystems. Theobjectiveofthismonographistostudystructure-preservingalgorithmsfor oscillatoryproblemsthatariseinawiderangeoffieldssuchasastronomy,molecu- lardynamics,classicalmechanics,quantummechanics,chemistry,biologyanden- gineering.Suchproblemscanoftenbemodeledbyinitialvalueproblemsofsecond- order differential equations with a linear term characterizing the oscillatory struc- tureofthesystems.Sincegeneral-purposehighorderRunge–Kutta(RK)methods, Runge–Kutta–Nyström(RKN)methods,andlinearmultistepmethods(LMM)can- not respect the special structures of oscillatory problems in long-term integration, innovativeintegratorshavetobedesigned.Thismonographsystematicallydevelops theories and methods for solving second-order differential equations with oscilla- torysolutions. As the basis of the whole monograph, Chap. 1 reviews the general notions and ideasrelatedtothenumericalintegrationofoscillatorydifferentialequations.Chap- ter2presentsmultidimensionalRKNmethodsadaptedtosecond-orderoscillatory systems. Chapter 3 proposes extended Runge–Kutta–Nyström (ERKN) methods for ini- tial value problems of second-order oscillatory systems with a constant frequency matrix or with a variable frequency matrix. The scheme of ERKN methods incor- v vi Preface porates the particular structure of the differential equations into both the internal stagesandtheupdates.Atri-coloredtreetheory,namely,thespecialextendedNys- tröm tree (SEN-tree) theory and the related B-series theory are established, based onwhichtheorderconditionsforERKNmethodsarederived.Therelationbetween ERKNmethodsandexponentiallyfittedmethodsisinvestigated.Multidimensional ERKNmethodsandmultidimensionalexponentiallyfittedmethodsareconstructed. Chapter4focusesonERKNmethodsforoscillatoryHamiltoniansystems.The symplecticityandsymmetryconditionsforERKNmethodsarepresented.Symplec- ticandsymmetricERKN(SSERKN)methodsareappliedtotheFermi–Pasta–Ulam problemandsomenonlinearwaveequationssuchasthesine-Gordonequation. TheideaofERKNmethodsisextendedtotwo-stephybridmethodsinChap.5, to Falkner-type methods in Chap. 6, to energy-preserving methods in Chap. 7, to asymptotic methods for highly oscillatory problems in Chap. 8, and to multi- symplecticmethodsforHamiltonianpartialdifferentialequationsinChap.9. All the numerical integrators presented in this monograph have been tested for oscillatory problems from a variety of applications. They are shown to be more efficientthansomeexistinghighqualitymethodsinthescientificliterature. Chapters 1 and 2 and Sect. 3.1 of Chap. 3 are more theoretical. Scientists and engineers who are mainly interested in numerical integrators may skip them, and thiswillnotaffectthecomprehensionoftherestofthemonograph. We are grateful to all the friends and colleagues for their selfless help during the preparation of this monograph. Special thanks are due to John Butcher of The University of Auckland, Christian Lubich of Universität Tübingen, Arieh Iserles of University of Cambridge, Jeff Cash of Imperial College London, Maarten de HoopofPurdueUniversity,QinShengofBaylorUniversity,TobiasJahnkeofKarl- sruher Institut für Technologie (KIT), Achim Schädle of Heinrich Heine Univer- sityDüsseldorf,ReinoutQuispelandDavidMcLarenofLaTrobeUniversity,Jesus Vigo-AguiarofUniversidaddeSalamanca,andRichardTerrillofMinnesotaState Universityfortheirencouragement. Wearealsogratefultomanyfriendsandcolleaguesforreadingthemanuscript andfortheirvaluablesuggestionsanddiscussions.Inparticular,wearegratefulto RobertPengKongChanofTheUniversityofAuckland,WeixingZheng,ZuheShen ofNanjingUniversity,JianlinXiaofPurdueUniversity,AdrianTurtonHillofBath University, Jichun Li of University of Nevada, Las Vegas, and Xiaowen Chang of McGillUniversity. Thanksalsogotothefollowingpeoplefortheirvarioushelpandsupport:Cheng Fang,PeihengWu,RongZhang,CongCong,ManchunLi,ShengwangWang,Jipu Ma,QiguangWu,XianglinFei,LinLiu,YuchengSu,XuesongBao,ChengsenLin, WentingTong,ChunhongXie,DongpingJiang,ZixiangOuyang,LiangshengLuo, Jinxi Zhao, Xinbao Ning, Weixue Shi, Chengkui Zhong, Jiangong You, Hourong Qin, Huicheng Yin, Xiaosheng Zhu, Zhiwei Sun, Qiang Zhang, Gaofei Zhang, ChunLiandZhiQianofNanjingUniversity,YaolinJiangofXi’anJiaoTongUni- versity, Yongzhong Song and Yushun Wang of Nanjing Normal University, Jialin Hong and Zaijiu Shang of Chinese Academyof Sciences, Jijun Liu and Zhizhong Sun of Southeast University, Shoufo Li and Aiguo Xiao of Xiang Tan University, Preface vii Chuanmiao Chen of Hunan Normal University, Siqing Gan of Central South Uni- versity, Chengjian Zhang and Chengming Huang of Huazhong University of Sci- ence&Technology,ShuanghuWangoftheInstituteofAppliedPhysicsandCom- putational Mathematics, Beijing, Hongjiong Tian of Shanghai Normal University, Yongkui Zou of Jilin University, Jingjun Zhao of Harbin Institute of Technology, Qinghong Li of Chuzhou University, Yonglei Fang of Zaozhuang University, Fan YangandHongliYangofNanjingInstituteofTechnology,JiyongLiofHebeiNor- malUniversity,WeiShi,KaiLiu,QihuaHuang,JunWu,JinsongYuandGuozhong Hu. We would like to thank Ji Luo for her help with the editing, the editorial and productiongroupoftheSciencePress,Beijing,andSpringer-Verlag,Heidelberg. Wealsothankourfamiliesfortheirloveandsupportthroughoutalltheseyears. The work on this monograph was supported in part by the Specialized Re- search Foundation for the Doctoral Program of Higher Education under Grant 20100091110033, by the 985 Project at Nanjing University under Grant 9112020301, by the Natural Science Foundation of China under Grant 10771099, Grant11271186and Grant11171155,andby thePriority AcademicProgramDe- velopmentofJiangsuHigherEducationInstitutions. Nanjing XinyuanWu XiongYou BinWang Contents 1 Runge–Kutta(–Nyström) Methods for OscillatoryDifferential Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1.1 RKMethods,RootedTrees,B-SeriesandOrderConditions . . . . 1 1.2 RKNMethods,NyströmTreesandOrderConditions. . . . . . . . 8 1.2.1 FormulationoftheScheme . . . . . . . . . . . . . . . . . 8 1.2.2 NyströmTreesandOrderConditions . . . . . . . . . . . . 9 1.2.3 TheSpecialCaseinAbsenceoftheDerivative . . . . . . . 18 1.3 DispersionandDissipationofRK(N)Methods . . . . . . . . . . . 19 1.3.1 RKMethods . . . . . . . . . . . . . . . . . . . . . . . . . 20 1.3.2 RKNMethods . . . . . . . . . . . . . . . . . . . . . . . . 21 1.4 SymplecticMethodsforHamiltonianSystems . . . . . . . . . . . 22 1.5 CommentsonStructure-PreservingAlgorithmsforOscillatory Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24 2 ARKNMethods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27 2.1 TraditionalARKNMethods . . . . . . . . . . . . . . . . . . . . . 27 2.1.1 FormulationoftheScheme . . . . . . . . . . . . . . . . . 28 2.1.2 OrderConditions . . . . . . . . . . . . . . . . . . . . . . 29 2.2 SymplecticARKNMethods . . . . . . . . . . . . . . . . . . . . . 32 2.2.1 SymplecticityConditionsforARKNIntegrators . . . . . . 33 2.2.2 ExistenceofSymplecticARKNIntegrators . . . . . . . . . 37 2.2.3 PhaseandStabilityPropertiesofMethodSARKN1s2 . . . 41 2.2.4 NonexistenceofSymmetricARKNMethods . . . . . . . . 43 2.2.5 NumericalExperiments . . . . . . . . . . . . . . . . . . . 44 2.3 MultidimensionalARKNMethods . . . . . . . . . . . . . . . . . 48 2.3.1 FormulationoftheScheme . . . . . . . . . . . . . . . . . 48 2.3.2 OrderConditions . . . . . . . . . . . . . . . . . . . . . . 52 2.3.3 PracticalMultidimensionalARKNMethods . . . . . . . . 57 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60 ix x Contents 3 ERKNMethods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63 3.1 ERKNMethods . . . . . . . . . . . . . . . . . . . . . . . . . . . 63 3.1.1 FormulationofMultidimensionalERKNMethods . . . . . 64 3.1.2 SpecialExtendedNyströmTreeTheory. . . . . . . . . . . 65 3.1.3 OrderConditions . . . . . . . . . . . . . . . . . . . . . . 73 3.2 EFRKNMethodsandERKNMethods . . . . . . . . . . . . . . . 76 3.2.1 One-DimensionalCase . . . . . . . . . . . . . . . . . . . 77 3.2.2 MultidimensionalCase . . . . . . . . . . . . . . . . . . . 80 3.3 ERKN Methods for Second-Order Systems with Variable PrincipalFrequencyMatrix . . . . . . . . . . . . . . . . . . . . . 82 3.3.1 AnalysisThroughanEquivalentSystem . . . . . . . . . . 82 3.3.2 TowardsERKNMethods . . . . . . . . . . . . . . . . . . 83 3.3.3 NumericalIllustrations . . . . . . . . . . . . . . . . . . . 86 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88 4 SymplecticandSymmetricMultidimensionalERKNMethods . . . . 91 4.1 SymplecticityandSymmetryConditionsforMultidimensional ERKNIntegrators . . . . . . . . . . . . . . . . . . . . . . . . . . 91 4.1.1 SymmetryConditions . . . . . . . . . . . . . . . . . . . . 92 4.1.2 SymplecticityConditions . . . . . . . . . . . . . . . . . . 94 4.2 ConstructionofExplicitSSMERKNIntegrators . . . . . . . . . . 99 4.2.1 TwoTwo-StageSSMERKNIntegratorsofOrderTwo . . . 99 4.2.2 AThree-StageSSMERKNIntegratorofOrderFour . . . . 101 4.2.3 StabilityandPhasePropertiesofSSMERKNIntegrators . . 103 4.3 NumericalExperiments . . . . . . . . . . . . . . . . . . . . . . . 105 4.4 ERKNMethodsforLong-TermIntegrationofOrbitalProblems . . 111 4.5 SymplecticERKNMethodsforTime-DependentSecond-Order Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111 4.5.1 Equivalent Extended Autonomous Systems for Non- autonomousSystems . . . . . . . . . . . . . . . . . . . . 112 4.5.2 Symplectic ERKN Methods for Time-Dependent HamiltonianSystems . . . . . . . . . . . . . . . . . . . . 113 4.6 ConcludingRemarks. . . . . . . . . . . . . . . . . . . . . . . . . 115 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117 5 Two-StepMultidimensionalERKNMethods . . . . . . . . . . . . . . 121 5.1 TheScheifeleTwo-StepMethods . . . . . . . . . . . . . . . . . . 121 5.2 FormulationofTSERKNMethods . . . . . . . . . . . . . . . . . 124 5.3 OrderConditions. . . . . . . . . . . . . . . . . . . . . . . . . . . 127 5.3.1 B-SeriesonSENT . . . . . . . . . . . . . . . . . . . . . . 127 5.3.2 One-StepFormulation . . . . . . . . . . . . . . . . . . . . 132 5.3.3 OrderConditions . . . . . . . . . . . . . . . . . . . . . . 133 5.4 ConstructionofExplicitTSERKNMethods . . . . . . . . . . . . 136 5.4.1 AMethodwithTwoFunctionEvaluationsperStep. . . . . 136 5.4.2 MethodswithThreeFunctionEvaluationsperStep. . . . . 138 5.5 StabilityandPhasePropertiesoftheTSERKNMethods . . . . . . 142 Contents xi 5.6 NumericalExperiments . . . . . . . . . . . . . . . . . . . . . . . 145 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 148 6 AdaptedFalkner-TypeMethods . . . . . . . . . . . . . . . . . . . . . 151 6.1 Falkner’sMethods . . . . . . . . . . . . . . . . . . . . . . . . . . 151 6.2 FormulationoftheAdaptedFalkner-TypeMethods. . . . . . . . . 152 6.3 ErrorAnalysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . 156 6.4 Stability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 161 6.5 NumericalExperiments . . . . . . . . . . . . . . . . . . . . . . . 166 AppendixA DerivationofGeneratingFunctions(6.14)and(6.15) . . . 169 AppendixB Proofof(6.24) . . . . . . . . . . . . . . . . . . . . . . . 170 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 171 7 Energy-PreservingERKNMethods . . . . . . . . . . . . . . . . . . . 173 7.1 TheAverage-Vector-FieldMethod . . . . . . . . . . . . . . . . . 173 7.2 Energy-PreservingERKNMethods . . . . . . . . . . . . . . . . . 175 7.2.1 FormulationoftheAAVFmethods . . . . . . . . . . . . . 175 7.2.2 AHighlyAccurateEnergy-PreservingIntegrator . . . . . . 176 7.2.3 TwoPropertiesoftheIntegratorAAVF-GL . . . . . . . . . 178 7.3 NumericalExperimentontheFermi–Pasta–UlamProblem. . . . . 179 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 183 8 EffectiveMethodsforHighlyOscillatorySecond-OrderNonlinear DifferentialEquations . . . . . . . . . . . . . . . . . . . . . . . . . . 185 8.1 NumericalConsiderationofHighlyOscillatorySecond-Order DifferentialEquations . . . . . . . . . . . . . . . . . . . . . . . . 185 8.2 TheAsymptoticMethodforLinearSystems . . . . . . . . . . . . 187 8.3 WaveformRelaxation(WR)MethodsforNonlinearSystems . . . 190 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 195 9 ExtendedLeap-FrogMethodsforHamiltonianWaveEquations . . . 197 9.1 ConservationLawsandMulti-SymplecticStructuresofWave Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 197 9.1.1 Multi-SymplecticConservationLaws . . . . . . . . . . . . 197 9.1.2 ConservationLawsforWaveEquations . . . . . . . . . . . 199 9.2 ERKNDiscretizationofWaveEquations . . . . . . . . . . . . . . 200 9.2.1 Multi-SymplecticIntegrators . . . . . . . . . . . . . . . . 200 9.2.2 Multi-SymplecticExtendedRKNDiscretization . . . . . . 200 9.3 ExplicitExtendedLeap-FrogMethods . . . . . . . . . . . . . . . 208 9.3.1 Eleap-FrogI:AnExplicitMulti-SymplecticERKN Scheme. . . . . . . . . . . . . . . . . . . . . . . . . . . . 208 9.3.2 Eleap-FrogII:AnExplicitMulti-SymplecticERKN-PRK Scheme. . . . . . . . . . . . . . . . . . . . . . . . . . . . 215 9.3.3 AnalysisofLinearStability . . . . . . . . . . . . . . . . . 215 9.4 NumericalExperiments . . . . . . . . . . . . . . . . . . . . . . . 217 9.4.1 TheConservationLawsandtheSolution . . . . . . . . . . 217 9.4.2 DispersionAnalysis . . . . . . . . . . . . . . . . . . . . . 224 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 229