Table Of ContentModelingandSimulationinScience,EngineeringandTechnology
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Modeling and
Computational Methods
for Kinetic Equations
Pierre Degond
Lorenzo Pareschi
Giovanni Russo
Editors
Springer Science+Business Media, LLC
Pierre Degond Lorenzo Pareschi
Universite Paul Sabatier Universita di Ferrara
Department of Mathematics Department of Mathematics
31 062 Toulouse Cedex 1-44100 Ferrara
France Italy
Giovanni Russo
Universita di Catania
95125 Catania
Italy
Library of Congress Cataloging-in-Publication Data
Modeling and computational methods for kinetic equations / Pierre Oegond, Lorenzo
Pareschi, Giovanni Russo, editors.
p. cm. - (Modeling and simulation in science, engineering & technology)
Inc1udes bibliographical references.
ISBN 978-1-4612-6487-3 ISBN 978-0-8176-8200-2 (eBook)
DOI 10.1007/978-0-8176-8200-2
1. Kinetic theory of matter. 2. Oegond, Pierre. II. Pareschi, Lorenzo. III. Russo,
Giovanni. IV. Series.
QC174.9.M632003
530.13'6-dc22 2003063694
CIP
AMS Subject Classifications: 65C05, 65M06, 65M70, 76P05, 76X05, 76Y05, 82840, 82C22, 82C40,
82C70, 82005, 82DIO, 82037, 90820
ISBN 978-1-4612-6487-3 Printed on acid-free paper.
© 2004 Springer Science+Business Media New York
Originally published by Birkhăuser Boston in 2004
Softcover reprint ofthe hardcover Ist edition 2004
Ali rights reserved. This work may not be translated or copied in whole or in part without the written
permission of the publisher Science+Business Media, LLC
except for brief excerpts in connection with reviews or
scholarly analysis. Use in connection with any form of information storage and retrieval, electronic
adaptation, computer software, or by similar or dissimilar methodology now known or hereafter
developed is forbidden.
The use in this publication of trade names, trademarks, service marks and similar terms, even if they
are not identified as such, is not to be taken as an expression of opinion as to whether or not they are
subject to property rights.
987654321 SPIN 10934692
www.birkhasuer-science.com
Contents
Preface ......................................................... vii
PartI RarefiedGases
1.Macroscopiclimits ofthe Boltzmannequation: a review
PierreDegond.................................................... 3
2.Momentequationsforchargedparticles: global existenceresults
GiuseppeAli, AngeloMarcello Anile .................................. 59
3.Monte-Carlomethodsfor the Boltzmannequation
SergejRjasanow " 81
4.Accurate numerical methods for the Boltzmannequation
Francis Filbet, GiovanniRusso 117
5.Finite-differencemethodsfor the Boltzmannequationfor binarygas
mixtures
KazuoAoki, Shingo Kosuge 147
PartII Applications
6.Plasmakinetic models: the Fokker-Planck-Landauequation
LaurentDesvillettes 171
7.On multipoleapproximationsofthe Fokker-Planck-Landauoperator
MohammedLemou ................................................ 195
8.Trafficflow:modelsand numerics
Axel Klar, RaimundWegener 219
vi Contents
9.Modelling and numerical methods for granulargases
LorenzoPareschi, GiuseppeToscani 259
10.Quantumkinetic theory: modelling and numericsfor Bose-Einstein
condensation
Weizhu Bao,LorenzoPareschi,PeterA.Markowich 287
11.On coalescenceequationsand related models
PhilippeLaurencot, StephaneMischler 321
Preface
The aim of this book is to provide a general overview of kinetic models and their
applications to various contexts (gas dynamics, semiconductor modelling, granular
flows, traffic flows, and so on.) Particular emphasis will be given to the derivation
ofthe models and to the modem numerical methods available to obtain quantitative
predictionsfrom the models.The mathematicaltreatment,although rigorous, will be
mostlymaintainedatalevelaccessibletoabroadrange ofreaders, includinggraduate
studentsinappliedsciencesand engineers.
The relation between kinetic models and simpler, macroscopic models will be
addressed in most chapters. This aspect is very importantinmany applications,and
exposes the readertothechallengeofmultiscale modelling.
Thebookisdividedintwoparts.Thefirstismainlydevotedtothemostfundamen
talkinetic model:the Boltzmannequation of rarefied gas dynamics. Its connections
with macroscopicmodels through hydrodynamiclimits and momentsclosurehierar
chies aredeveloped, astheyplay importantroles inthe morecommondescriptionof
gasesandfluids.Then,themostwidely usednumericalmethods forthediscretization
ofthe Boltzmannequationare reviewed:theMonte-Carlomethod, spectralmethods
and finite-difference methods. The second part is devoted to more specific applica
tions:plasmakinetic models withtheFokker-Planck-Landauequationanditsnumer
ical discretization, traffic flow modelling, granular media, quantum kinetic models
andcoagulation-fragmentationproblems.
Ineach case, both modelling aspects and numerical methods are discussed. The
originalityofthisbookisintheconsistenttreatmentofthemodels, bothfromthepoint
ofview oftheory and modelling and from that ofthe numerical discretization. Most
oftheexistingmonographsfocusoneitheroneortheotherofthesetwoaspects.How
ever,bringing these two aspects togethershines light on points which are important
butwhichareverylikelytobediscardedinmorefocused approaches.Forinstance,the
developmentofspectral ormultipolemethods forkinetic equationswasmotivatedby
thesearch forefficient waysofdiscretizingthe Boltzmann operatorwhile preserving
anaccuratedescriptionofthevariousconservationlawsaswellasentropydissipation.
These properties are deeply related tothe kind ofsystem the model aims atdescrib
ing.The same considerations are obviously true for trafficflow modelling,granular
viii Preface
media, quantum kinetic models, or coagulation-fragmentation problems, which are
the fourspecific applications thepresent book intends todevelop.
At the end ofeach chapter, alist ofreferences will address the interestedreader
towardmoredetailed treatmentofthesubject,andinparticulartosomeoftheresearch
trends inapplied kinetic theory.
Thefirstchapterofthebookisanintroductiontothekineticdescriptionofparticle
dynamics.After an overview of available models for particle dynamics, the Boltz
mann equationofrarefied gases isderived, anditsmain mathematicalproperties are
recalled. Particular emphasis is given to the derivation ofthe hydrodynamical lim
its, such as Euler and Navier-Stokes equations, by the formal procedure of Hilbert
andChapman-Enskogexpansion,whichconnectthemicroscopicworldofmolecules
with the macroscopic world ofgases, fluids, andthermodynamics.
The second chapterillustratesthemathematical propertiesofaclass ofhydrody
namical models that describecharge transport insemiconductors. Starting from the
semiclassical Boltzmann equation for semiconductors, hydrodynamical models are
deduced by applyingaprocedure introducedby Levermore, andbased onthe maxi
mumentropyprinciple.Itisshownthat,becauseofthespecialstructureofmomentum
space in acrystal (consequence ofperiodicity and band structure of the lattice), the
hydrodynamicalmodelsofsemiconductorspossesspeculiarproperties,differentfrom
thoseofmoment-basedmethods ingases.Inparticular, alocalexistenceresult anda
global existenceresult ofsmooth solutions around equilibriumarepresented.
The next three chapters are devoted to apresentationof the main tools available
for numerical approximationoftheBoltzmann equation.
Several challenges are encountered in the construction of effective numerical
schemes for kinetic equations because of the dimensionality of the problem (the
density function depends onsevenindependentscalar variables: time,physicalspace
and velocity space); thenonlocal nature (invelocity) ofthecollisionalkernel, which
makes it hard to compute it efficiently; the nonlinearity of the problem; and the
requirement to maintain the conservation properties of the equation at a discrete
level.The various schemes take these requirements into accountand satisfy them at
different levels.
Thefirst,andprobablymorewidely used,toolistheMonte-Carlomethod,which
hasseveraladvantagesoverpresentdaydeterministicschemes.Itisveryefficient,be
ingthemethodwiththelowestcomputationalcostperdiscrete degreeoffreedom, and
itisextremelyrobust,beingabletotreatawiderangeofregimes, includingsituations
that are very far from thermodynamical equilibrium. Furthermore,the Monte-Carlo
methodcan begeneralizedtoincludealargenumberofphysicaleffects.This chapter
isself-containedandcan beread independentlyofthe others.
Chapter4 describes a deterministic numerical method that can be used when a
highlyaccuratesolutionisrequired, andthesystemisnottoofarfromthermodynami
calequilibrium.Deterministicmethods areusuallymoreexpensivethanMonte-Carlo
methods, but they can provide more accurate solutions, without the statistical noise
typical oftheMonte-Carloapproach.Themethod presentedinthechapterisahybrid
method that combines a third-order accurate discretization in space with spectrally
Preface ix
accuratediscretizationinvelocity. Effectivetimeevolutionschemesallowgoodeffi
ciencyeveninregimesclosetothehydrodynamicallimit.
Thenextchapterillustratesthemainfeaturesoffinite differencediscretizationof
theBoltzmannequationanditsapplication togasmixtures.Amongcommonlyused
deterministicschemes,thefinitedifferenceschemesproposedbytheJapaneseschool
areprobablythe most wellestablished.Althoughnotquiteas accurateas spectral
methods,theyaremoreflexible,andallowvelocitydiscretization,whichcanbemore
easilyfittedtotheproblem.
Thesecondpartofthebookisdevotedtoapplications ofkineticequations.Far
frombeingexhaustive,thispartgivesabroadviewoftheuseofthekineticapproach
toseveraldifferentcontexts.
Chapters6and7dealwiththeFokker-Planck-Landauequationofplasmaphysics.
Theequationprovidesakineticdescriptionoftheevolutionofchargedparticles(elec
tronsandions),andarisesnaturallywhentheinteractionsamongtheparticlespro
ducemanysmalldeflections,andfewlargeanglescatters.ThechapterbyDesvillettes
iscenteredonthemathematical propertiesoftheFokker-Planck-Landauequation,
whilethechapterbyLemoudescribesapowerful methodforthenumericalsolution
oftheequation.
Chapter8isaself-containedreviewarticleontrafficflowmodelling.Bothkinetic
andhydrodynamical modelsare presented, and particularemphasisisgivento the
numerical techniquesusedfortheapproximatesolutionoftheequations.
Thenextchapterisabriefreviewofmodernkineticmodelsofgranularmaterial.
Inviewofthe numerousindustrialapplications, granularflow hasattractedalotof
attentioninrecentyears.Theinelasticityofgraincollisionscreatesnewinteresting
mathematical problems,whichare notpresentin thestandardBoltzmannequation
ofgasdynamics.Aftertheintroduction ofcertainkineticmodelsofdilutegranular
systems,thecoolingprocessofthesystemisstudiedandsomehydrodynamicalmod
els are derived.Accuratenumerical methodsbasedon a spectralrepresentation in
velocityarealsopresented,andthedevelopmentoffastalgorithmsisconsidered.
Chapter 10reviewssomemodellingandnumerical aspectsin quantumkinetic
theoryfora gas of interactingbosons(theso-calledBose-Einstein condensation).
Particularcare is devotedto the development of efficient numericalschemes for
the quantum Boltzmannequation that preservethe main physicalfeatures of the
continuousproblem,namelyconservationofmassandenergy,theentropyinequality,
and generalizedBose-Einstein distributions as steadystates. These propertiesare
essentialinordertodevelopnumericalmethodsthatareabletocapturethechallenging
phenomenon ofbosoncondensation.
Thelastchapterdealswiththemathematicaldescriptionofcoagulationphenom
ena.Theaimofthechapteristopresentanoverviewofthemathematical analysisof
coalescenceequationsandrelatedmodels,withafocusonthestatisticaldescription
atthekineticlevel.Someofthemainmathematical problemsandresultswithphysi-
x Preface
cal interest are presented, together with mathematical tools and strategies useful to
further investigate thesemodels.
Toulouse PierreDegond
March 2004 LorenzoPareschi
GiovanniRusso
