Modeling and Computational Methods for Kinetic Equations PDF
Preview Modeling and Computational Methods for Kinetic Equations
ModelingandSimulationinScience,EngineeringandTechnology SeriesEditor NicolaBellomo PolitecnicodiTorino Italy AdvisoryEditorialBoard M.Avellaneda(ModelinginEconomics) H.G.Othmer(MathematicalBiology) CourantInstituteofMathematicalSciences DepartmentofMathematics NewYorkUniversity UniversityofMinnesota 251MercerStreet 270AVincentHall NewYork,NY10012,USA Minneapolis,MN55455,USA avellaneda~cims.nyu.edu othmer~ath.umn.edu K.J.Bathe(SolidMechanics) L.Preziosi(IndustrialMathematics) DepartmentofMechanicalEngineering DipartimentodiMatematica MassachusettsInstituteofTechnology PolitecnicodiTorino Cambridge,MA02139,USA CorsoDucadegliAbruzzi24 kjb~it.edu 10129Torino,Italy preziosi~polito.it P.Degood(Semiconductor&TransportModeling) Math6matiquespourrlndustrieetlaPhysique V.Protopopescu(CompetitiveSystems,Epistemology) Universit6P.SabatierToulouse3 CSMD 118RoutedeNarbonne OakRidgeNationalLaboratory 31062ToulouseCedex,France OakRidge,TN3783Hl363,USA degond~ip.ups-tlse.fr vvp~epmnas.epm.ornl.gov MAHerreroGarria(MathematicalMethods) K.R.Rajagopa/(MultiphaseFlows) DepartamentodeMatematicaAplicada DepartmentofMechanicalEngineering UniversidadComplutensedeMadrid TexasMMUniversity AvenidaComplutensesin CollegeStation,TX77843,USA 28040Madrid,Spain KRajagopal~engr .tamu.edu herrero~sunma4.mat.ucm.es Y.Sooe(FluidDynamicsinEngineeringSciences) W.K1iemann(StochasticModeling) ProfessorEmeritus DepartmentofMathematics KyotoUniversity IowaStateUniversity 23D-133Iwakura-Nagatani-dlo 400CarverHall SakylrkuKyoto606-0026,Japan Ames,IA50011,USA sone~yoshio.mbox.media.kyoto-u.ac.jp kliemann~iastate.edu 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
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