Table Of ContentLecture 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
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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
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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
