Two-Scale Approach to Oscillatory Singularly Perturbed Transport Equations PDF
Preview Two-Scale Approach to Oscillatory Singularly Perturbed Transport Equations
Lecture Notes in Mathematics 2190 Emmanuel Frénod Two-Scale Approach to Oscillatory Singularly Perturbed Transport Equations Lecture Notes in Mathematics 2190 Editors-in-Chief: Jean-MichelMorel,Cachan BernardTeissier,Paris AdvisoryBoard: MichelBrion,Grenoble CamilloDeLellis,Zurich AlessioFigalli,Zurich DavarKhoshnevisan,SaltLakeCity IoannisKontoyiannis,Athens GáborLugosi,Barcelona MarkPodolskij,Aarhus SylviaSerfaty,NewYork AnnaWienhard,Heidelberg Moreinformationaboutthisseriesathttp://www.springer.com/series/304 Emmanuel Frénod Two-Scale Approach to Oscillatory Singularly Perturbed Transport Equations 123 EmmanuelFrénod LMBA UniversitéBretagneSud Vannes,France ISSN0075-8434 ISSN1617-9692 (electronic) LectureNotesinMathematics ISBN978-3-319-64667-1 ISBN978-3-319-64668-8 (eBook) DOI10.1007/978-3-319-64668-8 LibraryofCongressControlNumber:2017950521 MathematicsSubjectClassification(2010):34Exx,35L02,65-xx,82Dxx ©SpringerInternationalPublishingAG2017 Thisworkissubjecttocopyright.AllrightsarereservedbythePublisher,whetherthewholeorpartof thematerialisconcerned,specificallytherightsoftranslation,reprinting,reuseofillustrations,recitation, broadcasting,reproductiononmicrofilmsorinanyotherphysicalway,andtransmissionorinformation storageandretrieval,electronicadaptation,computersoftware,orbysimilarordissimilarmethodology nowknownorhereafterdeveloped. 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Printedonacid-freepaper ThisSpringerimprintispublishedbySpringerNature TheregisteredcompanyisSpringerInternationalPublishingAG Theregisteredcompanyaddressis:Gewerbestrasse11,6330Cham,Switzerland Preface Two-scaleconvergenceisahomogenizationtool.Ihavechosentocompilelecture notesonthistopicforatleasttworeasons.First,two-scaleconvergenceiscertainly the homogenization tool that is easiest to handle. It can be conveniently used to tackle many phenomena involving oscillations or heterogeneities. With two- scale convergence, we can design effective models in a constructive way and without much analytical material. Besides, the effective models that are based on two-scale convergence do not generate (or explicitly contain) the oscillations or heterogeneities of the studied phenomena, but only their average effects. Yet, two-scale convergence is a genuine homogenization method in the sense that the methods leading to the effective models rely on similar routines than any other homogenization method. For instance, it uses estimates, the so-called weak formulationwithoscillatingtestfunctions,passingtothelimit,andmore.Therefore, I considertwo-scale convergencea goodstarting pointfor someonewhowants to diveintohomogenizationtheory. The second reason is that, based on the concept of two-scale convergence, it is possible to construct numerical methods for the simulation of phenomena that involve oscillations or heterogeneities. These methods enable us to simulate such phenomena with a high degree of accuracy, yet without requiring detailed input with respect to the underlying phenomenon. Moreover, the methods provide a sustainableapproachtoreconstructingthedetailsoftheoscillations.Theyseemto beparticularlysuitableforhyperbolicpartialdifferentialequations(PDEs)thatare singularlyperturbedby an oscillatory operator.In particular,they efficientlycarry out beam simulation and seem to have the capacity of tackling the simulation of phenomenathatoccurinsidetoTokamaksandstellarators. Of course, two-scale convergence is not a panacea. It comes with its own drawbacks. One main drawback is that it can only be adapted to problems that involve periodic oscillations with only one high frequency.Although the tool can be sharpened with the help of scale separation in order to work for problems that involve periodic oscillations with several isolated frequencies, it seems very difficulttoimproveitinsuchawaythatonecantackleproblemsinvolvingperiodic oscillationswitheventwofrequenciesofthesameorderofmagnitude. v vi Preface Let me emphasize, however,that this limitation of two-scale convergencedoes not imply that homogenizationis restricted to periodic oscillations with only one high frequency. There are many homogenization tools (that are harder to handle thantwo-scaleconvergence)thatcanbeusedtotackleproblemsinvolvingarbitrary oscillations, without restrictions on their periodicity or their isolation. For an introductionto those tools, I recommend the books by Tartar [66], Allaire [5], or CioranescuandDonato[16]. From a numerical point of view, the development of numerical methods that are based on homogenization tools (not only on two-scale convergence) in order to simulate complex systems involving oscillations or heterogeneities is a wide and essentially unexplored research field. Numerical methods of this kind could be an essential contribution for understanding a variety of problems involving several scales and oscillations or heterogeneities that are not at all understood nowadays.Amongthem,letmementionturbulence,thetransitionfromanatomic scaledescriptiontomesoscalemodelsandBlochwavesinelectromagnetism. Forproblemsthatinvolvenopropagation,numericalmethodsthatarebasedon homogenizationhavebeendevelopedandstudiedindetail.ThebooksbyEfendiev andHou[22]andbyE[72]giveasoundintroductiontothosemethods(including the multiscale finite element method). Yet, the literature has far less to say on homogenization-based numerical methods for phenomena that include transport, particularly when strong oscillations are involved. I hope that the present lecture noteswillnarrowthisgap. Last but not least, let me say that the numericalaspects that I have introduced intheselecturenotesaremeantasanencouragementforthereadertoexplorethis fascinatingfieldofresearch. Vannes,France EmmanuelFrénod June5,2017 Acknowledgements The idea of writing this book emerged while I was preparing my lectures for Cemracs’11attheCIRM.IwouldthereforeliketothanktheorganizersofCemracs ’11forinvitingme: (cid:129) FrédéricCoquel(ÉcolePolytechnique,Palaiseau,France) (cid:129) MichaëlGutnic(UniversitédeStrasbourg,France) (cid:129) PhilippeHelluy(UniversitédeStrasbourg,France) (cid:129) FrédéricLagoutière(Universitéd’Orsay-Paris-Sud,France) AlargepartofthisbookwaswritteninShanghai(attheInstituteofNaturalSciences duringtheday,andin theSJTU cafeteriaatnight)duringmy stay in May2013.I wanttothankLeiZhangfromtheInstituteofNaturalScienceswhoinvitedmefor thisverypleasantandstudiousstay. IalsothanktheorganizersoftheMFNSchoolatÎledePorquerollesinJune2013 forinvitingmetogivelectures: (cid:129) WietzeHerremanandBérengèrePodvin(Universitéd’Orsay-Paris-Sud,France) (cid:129) AnneSergent(UniversitéPierreetMarieCurie,Paris,France) Finally, I would like to thank Ghouti Bereksi (Université Abou Bekr Belkaid, Tlemcen, Algeria) and Thao Thi Phong Ha (Université Bretagne Sud, Vannes, France) for proofreading the draft of this book and for pointing out misprints and inconsistencies. I trust that their help improved the quality of the manuscript considerably. My research activity is currently carried out within the framework of the EURO fusion Consortium (Project: CfP-WP14-ER-01/Swiss Confederation-01, CfP-WP14-ER-01/IPP-03&CfP-WP15-ER/IPP-01)andhasreceivedfundingfrom theEuratomResearchandTrainingProgramme2014–2018underthegrantagree- mentNo633053. vii Contents PartI Two-ScaleConvergence 1 Introduction .................................................................. 3 1.1 FirstStatementsonTwo-ScaleConvergence.......................... 3 1.2 Two-ScaleConvergenceandHomogenization ........................ 3 1.2.1 How Homogenization Led to the Concept ofTwo-ScaleConvergence.................................... 3 1.2.2 ARemarkConcerningPeriodicity............................ 15 1.2.3 ARemarkConcerningWeak-*Convergence................ 15 2 Two-ScaleConvergence:DefinitionandResults.......................... 21 2.1 BackgroundMaterialonTwo-ScaleConvergence..................... 21 2.1.1 Definitions..................................................... 21 2.1.2 LinkwithWeakConvergence................................. 23 2.2 Two-ScaleConvergenceCriteria....................................... 24 2.2.1 InjectionLemma .............................................. 24 2.2.2 Two-ScaleConvergenceCriterion............................ 29 2.2.3 StrongTwo-ScaleConvergenceCriterion.................... 30 3 Applications .................................................................. 35 3.1 HomogenizationofOrdinaryDifferentialEquations.................. 35 3.1.1 TextbookCase,SettingandAsymptoticExpansion......... 35 3.1.2 Justification of Asymptotic Expansion Using Two-ScaleConvergence....................................... 39 3.2 HomogenizationofOscillatorySingularly-PerturbedOrdinary DifferentialEquations .................................................. 43 3.2.1 EquationofInterestandSetting .............................. 43 3.2.2 AsymptoticExpansionResults ............................... 45 3.2.3 AsymptoticExpansionCalculations.......................... 47 3.2.4 JustificationUsingTwo-ScaleConvergenceI:Results...... 53 3.2.5 JustificationUsingTwo-ScaleConvergenceII:Proofs...... 54 ix x Contents 3.3 HomogenizationofHyperbolicPartialDifferentialEquations ....... 66 3.3.1 TextbookCaseandSetting.................................... 66 3.3.2 Order-0Homogenization...................................... 66 3.3.3 Order-1Homogenization...................................... 69 3.4 HomogenizationofLinearSingularly-PerturbedHyperbolic PartialDifferentialEquations........................................... 74 3.4.1 EquationofInterestandSetting .............................. 74 3.4.2 AnaPrioriEstimate........................................... 75 3.4.3 WeakFormulationwithOscillatingTestFunctions ......... 75 3.4.4 Order-0Homogenization:Constraint......................... 76 3.4.5 Order-0Homogenization:EquationforV.................... 76 3.4.6 Order-1Homogenization:Preparations:Equations forU andu..................................................... 78 3.4.7 Order-1 Homogenization:Strong Two-Scale Convergenceofu"............................................. 79 3.4.8 Order-1Homogenization:TheFunctionW1 ................. 80 3.4.9 Order-1 Homogenization:A Priori Estimate andConvergence .............................................. 82 3.4.10 Order-1Homogenization:Constraint......................... 83 3.4.11 Order-1Homogenization:EquationforV1................... 84 3.4.12 ConcerningNumerics......................................... 87 PartII Two-ScaleNumericalMethods 4 Introduction .................................................................. 91 5 Two-Scale NumericalMethodforthe Long-Term Forecast oftheDriftofObjectsinanOceanwithTideandWind................ 93 5.1 MotivationandModel .................................................. 93 5.1.1 Motivation ..................................................... 93 5.1.2 ModelofInterest.............................................. 94 5.2 Two-ScaleAsymptoticExpansion ..................................... 96 5.2.1 AsymptoticExpansion........................................ 96 5.2.2 Discussion ..................................................... 98 5.3 Two-ScaleNumericalMethod.......................................... 100 5.3.1 ConstructionoftheTwo-ScaleNumericalMethod.......... 100 5.3.2 ValidationoftheTwo-ScaleNumericalMethod............. 102 6 Two-ScaleNumericalMethodfortheSimulationofParticle BeamsinaFocussingChannel ............................................. 109 6.1 SomeWordsAboutBeamsandtheModelofInterest ................ 109 6.1.1 Beams.......................................................... 109 6.1.2 EquationsofInterest .......................................... 110 6.1.3 Two-ScaleConvergence....................................... 111
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