at ebookgate.com
Differencemethodsforsingularperturbation problems1stEditionGrigoryI.Shishkin
https://ebookgate.com/product/difference-methodsfor-singular-perturbation-problems-1st-editiongrigory-i-shishkin/
Download more ebook from https://ebookgate.com
More products digital (pdf, epub, mobi) instant download maybe you interests ...
Fitted Numerical Methods For Singular Perturbation Problems Error Estimates in the Maximum Norm for Linear Problems in One and Two Dimensions J J H Miller
https://ebookgate.com/product/fitted-numerical-methods-forsingular-perturbation-problems-error-estimates-in-the-maximumnorm-for-linear-problems-in-one-and-two-dimensions-j-j-h-miller/
Time Dependent Problems and Difference Methods 2nd Edition Bertil Gustafsson https://ebookgate.com/product/time-dependent-problems-anddifference-methods-2nd-edition-bertil-gustafsson/
Elliptic Mixed Transmission and Singular Crack Problems 4th Edition Gohar Harutyunyan
https://ebookgate.com/product/elliptic-mixed-transmission-andsingular-crack-problems-4th-edition-gohar-harutyunyan/
Perturbation Theory for Matrix Equations 1st Edition C.K. Chui
https://ebookgate.com/product/perturbation-theory-for-matrixequations-1st-edition-c-k-chui/
Variational Methods for Nonlocal Fractional Problems 1st Edition Giovanni Molica Bisci
https://ebookgate.com/product/variational-methods-for-nonlocalfractional-problems-1st-edition-giovanni-molica-bisci/
The stochastic perturbation method for computational mechanics 1st Edition Kaminski M.
https://ebookgate.com/product/the-stochastic-perturbation-methodfor-computational-mechanics-1st-edition-kaminski-m/
Bayesian Methods for Management and Business Pragmatic Solutions for Real Problems 1st Edition Eugene D. Hahn
https://ebookgate.com/product/bayesian-methods-for-managementand-business-pragmatic-solutions-for-real-problems-1st-editioneugene-d-hahn/
Dynamics of third order rational difference equations with open problems and conjectures 1st Edition Elias Camouzis
https://ebookgate.com/product/dynamics-of-third-order-rationaldifference-equations-with-open-problems-and-conjectures-1stedition-elias-camouzis/
Analytic Methods for Coagulation Fragmentation Models Volume I 1st Edition Jacek Banasiak (Author)
https://ebookgate.com/product/analytic-methods-for-coagulationfragmentation-models-volume-i-1st-edition-jacek-banasiak-author/
CHAPMAN
DIFFERENCE METHODS SINGULAR PERTURBATION PROBLEMS CHAPMAN & HALL/CRC Monographs and Surveys in Pure and Applied Mathematics Main Editors
H. Brezis, Université de Paris
R.G. Douglas, Texas A&M University
A. Jeffrey, University of Newcastle upon Tyne (Founding Editor)
Editorial Board
R. Aris, University of Minnesota
G.I. Barenblatt, University of California at Berkeley
H. Begehr, Freie Universität Berlin
P. Bullen, University of British Columbia
R.J. Elliott, University of Alberta
R.P. Gilbert, University of Delaware
R. Glowinski, University of Houston
D. Jerison, Massachusetts Institute of Technology
K. Kirchgässner, Universität Stuttgart
B. Lawson, State University of New York
B. Moodie, University of Alberta
L.E. Payne, Cornell University
D.B. Pearson, University of Hull
G.F. Roach, University of Strathclyde
I. Stakgold, University of Delaware
W.A. Strauss, Brown University
J. van der Hoek, University of Adelaide
DIFFERENCE METHODS FOR SINGULAR PERTURBATION PROBLEMS Grigory I. Shishkin Lidia P. Shishkina Chapman & Hall/CRC
Taylor & Francis Group
6000 Broken Sound Parkway NW, Suite 300 Boca Raton, FL 33487-2742
© 2009 by Taylor & Francis Group, LLC
Chapman & Hall/CRC is an imprint of Taylor & Francis Group, an Informa business
No claim to original U.S. Government works
Printed in the United States of America on acid-free paper 10 9 8 7 6 5 4 3 2 1
International Standard Book Number-13: 978-1-58488-459-0 (Hardcover)
This book contains information obtained from authentic and highly regarded sources. Reasonable efforts have been made to publish reliable data and information, but the author and publisher cannot assume responsibility for the validity of all materials or the consequences of their use. The authors and publishers have attempted to trace the copyright holders of all material reproduced in this publication and apologize to copyright holders if permission to publish in this form has not been obtained. If any copyright material has not been acknowledged please write and let us know so we may rectify in any future reprint.
Except as permitted under U.S. Copyright Law, no part of this book may be reprinted, reproduced, transmitted, or utilized in any form by any electronic, mechanical, or other means, now known or hereafter invented, including photocopying, microfilming, and recording, or in any information storage or retrieval system, without written permission from the publishers.
For permission to photocopy or use material electronically from this work, please access www.copyright.com (http://www.copyright.com/) or contact the Copyright Clearance Center, Inc. (CCC), 222 Rosewood Drive, Danvers, MA 01923, 978-750-8400. CCC is a not-for-profit organization that provides licenses and registration for a variety of users. For organizations that have been granted a photocopy license by the CCC, a separate system of payment has been arranged.
Trademark Notice: Product or corporate names may be trademarks or registered trademarks, and are used only for identification and explanation without intent to infringe.
Library of Congress Cataloging-in-Publication Data
Shishkin, G. I.
Difference methods for singular perturbation problems / Grigory I. Shishkin, Lidia P. Shishkina. p. cm. -- (Chapman & Hall/CRC monographs and surveys in pure and applied mathematics)
Includes bibliographical references and index. ISBN 978-1-58488-459-0 (hardback : alk. paper)
1. Singular perturbations (Mathematics) 2. Difference equations--Numerical solutions. 3. Algebra, Abstract. I. Shishkina, Lidia P. II. Title. III. Series.
QC20.7.P47S55 2008 515’.392--dc22 2008025636
Visit the Taylor & Francis Web site at http://www.taylorandfrancis.com and the CRC Press Web site at http://www.crcpress.com
Dedication Dedicatedtothememoryofacademicians AlexandrAndreevichSamarskiiand
NikolaiSergeevichBakhvalov
Preface xiii
IGridapproximationsofsingularperturbation partialdifferentialequations 1
1Introduction 3
1.1Thedevelopmentofnumericalmethodsforsingularly perturbedproblems......................3
1.2Theoreticalproblemsintheconstructionofdifferenceschemes6
1.3Themainprinciplesintheconstructionofspecialschemes8
1.4Moderntrendsinthedevelopmentofspecialdifference schemes.............................10
1.5Thecontentsofthepresentbook...............11
1.6Thepresentbook.......................12
1.7Theaudienceforthisbook..................16
2Boundaryvalueproblemsforellipticreaction-diffusion equationsindomainswithsmoothboundaries 17
2.1Problemformulation.Theaimoftheresearch.......17
2.2Estimatesofsolutionsandderivatives............19
2.3Conditionsensuring ε-uniformconvergenceofdifference schemesfortheproblemonaslab..............26
2.3.1Sufficientconditionsfor ε-uniformconvergenceof differenceschemes...................26
2.3.2Sufficientconditionsfor ε-uniformapproximationof theboundaryvalueproblem..............29
2.3.3Necessaryconditionsfordistributionofmeshpoints for ε-uniformconvergenceofdifferenceschemes. Constructionofcondensingmeshes..........33
2.4Monotonefinitedifferenceapproximationsoftheboundary valueproblemonaslab. ε-uniformlyconvergentdifference schemes.............................38
2.4.1Problemsonuniformmeshes.............38
2.4.2Problemsonpiecewise-uniformmeshes........44
2.4.3Consistentgridsonsubdomains............51
2.4.4 ε-uniformlyconvergentdifferenceschemes......57
2.5 Boundaryvalueproblemsindomainswithcurvilinear boundaries...........................58
2.5.1Adomain-decomposition-baseddifferenceschemefor theboundaryvalueproblemonaslab........58
2.5.2Adifferenceschemefortheboundaryvalueproblem inadomainwithcurvilinearboundary........67
3Boundaryvalueproblemsforellipticreaction-diffusion equationsindomainswithpiecewise-smoothboundaries75
3.1Problemformulation.Theaimoftheresearch.......75
3.2Estimatesofsolutionsandderivatives............76
3.3Sufficientconditionsfor ε-uniformconvergenceofadifference schemefortheproblemonaparallelepiped.........85
3.4Adifferenceschemefortheboundaryvalueproblemon aparallelepiped........................89
3.5Consistentgridsonsubdomains...............97
3.6Adifferenceschemefortheboundaryvalueproblemin adomainwithpiecewise-uniformboundary.........102
4Generalizationsforellipticreaction-diffusionequations109
4.1MonotonicityofcontinualanddiscreteSchwartzmethods.109
4.2Approximationofthesolutioninaboundedsubdomainfor theproblemonastrip.....................112
4.3Differenceschemesofimprovedaccuracyfortheproblemon aslab..............................120
4.4Domain-decompositionmethodforimprovediterative schemes.............................125
5Parabolicreaction-diffusionequations
133
5.1Problemformulation......................133
5.2Estimatesofsolutionsandderivatives............134
5.3 ε-uniformlyconvergentdifferenceschemes.........145
5.3.1Gridapproximationsoftheboundaryvalueproblem146
5.3.2Consistentgridsonaslab...............147
5.3.3Consistentgridsonaparallelepiped.........154
5.4Consistentgridsonsubdomains...............158
5.4.1Theproblemonaslab.................158
5.4.2Theproblemonaparallelepiped...........161
6Ellipticconvection-diffusionequations 165
6.1Problemformulation......................165
6.2Estimatesofsolutionsandderivatives............166
6.2.1Theproblemsolutiononaslab............166
6.2.2Theproblemonaparallelepiped...........169
6.3 Onconstructionof ε-uniformlyconvergentdifferenceschemes undertheirmonotonicitycondition..............176
6.3.1Analysisofnecessaryconditionsfor ε-uniform convergenceofdifferenceschemes...........177
6.3.2Theproblemonaslab.................180
6.3.3Theproblemonaparallelepiped...........183
6.4Monotone ε-uniformlyconvergentdifferenceschemes....185
7Parabolicconvection-diffusionequations 191
7.1Problemformulation......................191
7.2Estimatesoftheproblemsolutiononaslab.........192
7.3Estimatesoftheproblemsolutiononaparallelepiped...199
7.4Necessaryconditionsfor ε-uniformconvergenceof differenceschemes.......................206
7.5Sufficientconditionsfor ε-uniformconvergenceofmonotone differenceschemes.......................210
7.6Monotone ε-uniformlyconvergentdifferenceschemes....213
IIAdvancedtrendsin ε-uniformlyconvergent differencemethods 219
8Gridapproximationsofparabolicreaction-diffusion equationswiththreeperturbationparameters 221
8.1Introduction..........................221
8.2Problemformulation.Theaimoftheresearch.......222
8.3 Apriori estimates.......................224
8.4Gridapproximationsoftheinitial-boundaryvalueproblem230
9Applicationofwidthsforconstructionofdifference schemesforproblemswithmovingboundarylayers235
9.1Introduction..........................235
9.2Aboundaryvalueproblemforasingularlyperturbed parabolicreaction-diffusionequation.............237
9.2.1Problem(9.2),(9.1)..................237
9.2.2Somedefinitions....................238
9.2.3Theaimoftheresearch................240
9.3 Apriori estimates.......................241
9.4Classicalfinitedifferenceschemes..............243
9.5Constructionof ε-uniformandalmost ε-uniform approximationstosolutionsofproblem(9.2),(9.1).....246
9.6Differenceschemeonagridadaptedinthemoving boundarylayer.........................251
9.7Remarksandgeneralizations.................254
10
High-orderaccuratenumericalmethodsforsingularly perturbedproblems 259
10.1Introduction..........................259
10.2Boundaryvalueproblemsforsingularlyperturbedparabolic convection-diffusionequationswithsufficientlysmoothdata261
10.2.1Problemwithsufficientlysmoothdata........261
10.2.2Afinitedifferenceschemeonanarbitrarygrid....262
10.2.3Estimatesofsolutionsonuniformgrids.......263
10.2.4Special ε-uniformconvergentfinitedifferencescheme263
10.2.5Theaimoftheresearch................264
10.3 Apriori estimatesforproblemwithsufficientlysmoothdata265
10.4Thedefectcorrectionmethod.................266
10.5TheRichardsonextrapolationscheme............270
10.6Asymptoticconstructs.....................273
10.7Aschemewithimprovedconvergenceforfinitevaluesof ε .275
10.8Schemesbasedonasymptoticconstructs..........277
10.9Boundaryvalueproblemforsingularlyperturbedparabolic convection-diffusionequationwithpiecewise-smoothinitial data...............................280
10.9.1Problem(10.56)withpiecewise-smoothinitialdata.280
10.9.2Theaimoftheresearch................281
10.10 Apriori estimatesfortheboundaryvalueproblem(10.56) withpiecewise-smoothinitialdata..............282
10.11Classicalfinitedifferenceapproximations..........285
10.12Improvedfinitedifferencescheme...............287
11Afinitedifferenceschemeon apriori adaptedgridsfor asingularlyperturbedparabolicconvection-diffusion equation 289
11.1Introduction..........................289
11.2Problemformulation.Theaimoftheresearch.......290
11.3Gridapproximationsonlocallyrefinedgridsthatareuniform insubdomains.........................293
11.4Differenceschemeon apriori adaptedgrid.........297
11.5Convergenceofthedifferenceschemeon apriori adaptedgrid303
11.6Appendix............................307
12Onconditioningofdifferenceschemesandtheirmatrices forsingularlyperturbedproblems 309
12.1Introduction..........................309
12.2Conditioningofmatricestodifferenceschemesonpiecewiseuniformanduniformmeshes.ModelproblemforODE...311
12.3Conditioningofdifferenceschemesonuniformandpiecewiseuniformgridsforthemodelproblem.............316
12.4 Onconditioningofdifferenceschemesandtheirmatricesfor aparabolicproblem......................323
13Approximationofsystemsofsingularlyperturbedelliptic reaction-diffusionequationswithtwoparameters 327
13.1Introduction..........................327
13.2Problemformulation.Theaimoftheresearch.......328
13.3Compatibilityconditions.Some apriori estimates.....330
13.4Derivationof apriori estimatesfortheproblem(13.2)under thecondition(13.5)......................333
13.5 Apriori estimatesfortheproblem(13.2)undertheconditions (13.4),(13.6)..........................341
13.6Theclassicalfinitedifferencescheme.............343
13.7Thespecialfinitedifferencescheme.............345
13.8Generalizations.........................348
14Survey 349
14.1Applicationofspecialnumericalmethodstomathematical modelingproblems.......................349
14.2Numericalmethodsforproblemswithpiecewise-smoothand nonsmoothboundaryfunctions................351
14.3Ontheapproximationofsolutionsandderivatives.....352
14.4Ondifferenceschemesonadaptivemeshes..........354
14.5Onthedesignofconstructivedifferenceschemesforanelliptic convection-diffusionequationinanunboundeddomain..357
14.5.1Problemformulationinanunboundeddomain.The taskofcomputingthesolutioninaboundeddomain357
14.5.2Domainofessentialdependenceforsolutionsofthe boundaryvalueproblem................359
14.5.3Generalizations.....................363
14.6Compatibilityconditionsforaboundaryvalueproblemona rectangleforanellipticconvection-diffusionequationwitha perturbationvectorparameter................364
14.6.1Problemformulation..................365
14.6.2Compatibilityconditions................366
References 371
Preface The presentbookisdevotedtothe developmentofdifferenceschemes that converge ε-uniformly inthemaximumnormforarepresentativeclassof singularlyperturbedproblems.Italsodealswiththe justification oftheir convergence,and surveysnewdirectionsandapproachesdevelopedrecently,which areofimportanceforfurtherprogressinnumericalmethods.
ThebookwasintendedtobeanEnglishtranslationoftheRussianbook [138] ShishkinG.I.(1992). DiscreteApproximationsofSingularlyPerturbed EllipticandParabolicEquations.RussianAcademyofSciences,UralSection, Ekaterinburg(inRussian) thatwasinitiatedbyJohnJ.H.Miller.ThetranslationwasmadebyZoraUzelac,butwedecidednottopublishthisversionof thebook.Theverydensenatureofthisbook,thatallowedustocoveralarge classofsingularlyperturbedboundaryvalueproblemsinlittlespace,wastoo difficultformostreadersandalsocreatedproblemsintheimplementationof theresults.Sincetheappearanceofthebook[138],newresultsandideas haveappearedthataredealtwithinthepresentbook.
First,Iwouldliketothankmyteachers.Myscientificinterestsincomputationalmathematicswereformedandmaturedundertheinfluenceofthe scientificschoolsoftheAcademiciansoftheRussianAcademyofScience. A.M.Il’in,A.A.Samarskii,N.S.Bakhvalov,G.I.Marchukandtheirinfluence ledtotheappearanceofmyseconddoctoralthesis.Thisthesisbecamethe basisof[138],andisacontinuinginfluenceonmywork.
ItiswithpleasurethatInotethelong-termandfruitfulcollaborationwith theIrishandDutchmathematiciansinthegroupsofJ.MillerandP.Hemker. Thiscollaborationbeganin1990,andyieldedprogressinthedevelopmentof numericalmethodsforproblemswithboundarylayers,andledtonewresults thatwerepublishedinnumerousjointpapersandintwobooks[87]and[33].
TheRussianscientistsK.V.Emelianov,V.D.Liseikin,P.N.Vabishchevich, V.B.Andreev,V.F.Butuzov,A.V.Gulin,I.G.Belukhina,N.V.Kopteva, V.V.Shaidurov,B.M.Bagaev,E.D.Karepova,M.M.Lavrentiev,Jr,Yu.M. Laevsky,A.I.Zadorin,A.D.LjashkoandI.B.Badrievalsoinfluencedmuchof thedetailoftheapproachesinitiatedin[138].
TheideatotranslateintoEnglishthebook[138]beganduringmycollaborationoverthelastdozenyearswithmathematiciansandtheirstudents, namely,J.J.H.Miller,E.O’Riordan,A.F.Hegarty,M.Stynes,A.Ansari(Ireland),P.W.Hemker,J.Maubach,P.Wesseling(theNetherlands),P.A.Farrell (USA),F.Lisbona,C.Clavero,J.L.Gracia,J.C.Jorge(Spain),D.Creamer, LinPin(Singapore),andthroughdiscussionsofpapers(basedonideasfrom
[138]) oninternationalconferenceswith,amongothers,I.P.Boglaev(New Zealand),R.E.O’Malley,R.B.Kellogg(USA),L.Tobiska,H.-G.Roos,G. Lube,T.Linß(Germany),WangSong(Australia),L.G.Vulkov,I.A.Braianov(Bulgaria),R. ˘ Ciegis(Lithuania),andP.P.Matus(Belarus).Numerous ideasfrom[138]wereextendedandpublishedinmanypapers.
MythanksespeciallytoL.P.Shishkina,mybetterhalf,andmainassistantcolleagueandmathematicianforparticipationasco-authorinwritingthis book,forenormousscientificandtechnicalsupport.Shehaspreparedthe presentbookincludingallstages:theclarificationofresultsbynumerous discussions,preparationinLaTeX,thetranslation,compilingtheIndex,and reviewingthepage-proofs.
SignificantassistanceinthepreparationoftheEnglishversionofthepresent book,inthetranslationfromRussian-EnglishtoidiomaticEnglish,wasmade byM.Stynes(PartI,andfragmentsof PartII)andM.Mortell(thePreface andtheIntroduction)towhomIwouldliketoexpressmydeepestthanks.
Mythankstoourassistant-colleagueI.V.Tselishchevaforsupportinthe processofpreparingthebook,participationinthetranslationofsomechapters fromPartII,oftheIntroduction,oftheSurvey,andmanyothertasks.
Iamgratefulforfinancialandmaterialsupport(scientificbooks,computationaltechnique)tothe
InstituteofMathematicsandMechanics,UralBranchoftheRussian AcademyofSciences,Yekaterinburg,Russia;
InstituteforNumericalComputationandAnalysis,Dublin(INCA), Ireland;
DepartmentofMathematicsatTrinityCollegeDublin,Ireland; CWI(ResearchInstituteoftheStichtingMathematischCentrum), Amsterdam,theNetherlands;
SchoolofMathematicalSciences,DublinCityUniversity,Ireland;
DepartmentofMathematicsandStatisticsattheUniversityof Limerick,Ireland;
NationalResearchInstituteforMathematicsandComputerScience(NUS),Singapore;
SchoolofMathematicalSciencesattheNationalUniversityofIrelandinCork(UCC),Ireland;
BooleCentreforResearchinInformaticsattheUCC,Ireland;
MathematicsApplicationsConsortiumforScienceandIndustry inIreland(MACSI)undertheScienceFoundationIreland(SFI) MathematicsInitiative; andtoChapman&Hallforfriendlycooperation.
In particular,theresearchworkofG.I.ShishkinandL.P.Shishkinawas supportedbythe
RussianFoundationforBasicResearchundergrants No.01-01-01022,04-01-00578,07-01-00729; grantRFBR-NWO(RFFI-NWO)04-01-89007-NWO a);
DutchResearchOrganisationNWOundergrants No.047.008.007,047.016.008.
TheresearchworkofG.I.Shishkinwasalsosupportedbythe InternationalCollaborationProgrammeofForbairt,Dublin,IrelandNo.IC/97/057; EnterpriseIrelandBasicResearchgrants SC–98–612,SC–2000–070.
G.I.Shishkin
PartI Gridapproximationsof singularperturbation partialdifferential equations Chapter 1 Introduction 1.1 The developmentofnumericalmethodsforsingularlyperturbedproblems
Thewideuseofcomputingtechniques,combinedwiththedemandsofscientificandtechnicalpractices,hasstimulatedthedevelopmentofnumerical methodstoagreatextent,andinparticular,methodsforsolvingdifferentialequations.Theefficiencyofsuchmethodsisgovernedbytheiraccuracy, simplicityincomputingthediscretesolutionandalsotheirrelativeinsensitivitytoparametersintheproblem.Atpresent,numericalmethodsforsolvingpartialdifferentialequations,inparticular,finitedifferenceschemes,are welldevelopedforwideclassesofboundaryvalueproblems(see,forexample, [79, 108, 100, 214, 91, 216]).
Amongboundaryvalueproblems,aconsiderableclassincludesproblemsfor singularlyperturbedequations,i.e.,differentialequationswhosehighest-order derivativesaremultipliedbya(perturbation)parameter ε.Theperturbation parameter ε maytakearbitraryvaluesintheopen-closedinterval(0, 1](see, e.g.,[211, 210, 57, 94, 62]).Solutionsofsingularlyperturbedproblems,unlike regularproblems,haveboundaryand/orinteriorlayers,thatis,narrowsubdomainsspecifiedbytheparameter ε onwhichthesolutionsvarybyafinite value.Thederivativesofthesolutioninthesesubdomainsgrowwithout boundas ε tendstozero.
Inthecaseofsingularlyperturbedproblems,theuseofnumericalmethods developedforsolvingregularproblemsleadstoerrorsinthesolutionthatdependonthevalueoftheparameter ε.Errorsofthenumericalsolutiondepend onthedistributionofmeshpointsandbecomesmallonlywhentheeffective mesh-sizeinthelayerismuchlessthanthevalueoftheparameter ε (see,e.g., [138, 87, 106, 33]).Suchnumericalmethodsturnouttobeinapplicablefor singularlyperturbedproblems.
Duetothis,thereisaninterestinthedevelopmentofspecialnumerical methodswheresolutionerrorsareindependentoftheparameter ε anddefinedonlybythenumberofnodesinthemeshesused,i.e., numericalmethods (inparticular, finitedifferenceschemes)that converge ε-uniformly.When thesolutionsbysuchmethodsare ε-uniformlyconvergent,wewillcallthese methodsandsolutions robust (asin[33]).Atpresent,onlyseveralbooks aredevotedtothe development ofnumericalmethodsforsolvingsingularly
perturbedproblems.Gridmethodsforboundaryvalueproblemsforpartial differentialequationsareconsideredinthebooks[138,87,33,75];seealso [26, 13, 14, 76]forordinarydifferentialequations.Inthebook[106],the authorsgiveanumberofresultsandalsoacomprehensivebibliographyon numericalmethodsforsolvingsingularlyperturbedproblemsforpartialdifferentialequationsandforordinarydifferentialequations.
ThepresentbookwasintendedtobeanEnglishtranslationofthebook [138].Avarietyofideasandapproachesfrom[138]havesincebeenfurther developed.Newapproachesandtrendsappear,whichrequirefurtherinvestigation.Inthepresentbook,weelaborateonapproachestothedevelopmentof ε-uniformlyconvergentnumericalmethodsforseveralboundaryvalueproblemsfrom[138]anddiscusssomenewtrendsinthedevelopmentofother methods,whichhaveappearedrecently.
Quiteoftensolutionsofboundaryvalueproblems,theirgridsolutions,and alsotheirconvergenceareconsideredusingmaximumnorms.Theuseofeither theenergynormor L1, L2-normsisinadequatetodescribethesolutionsof singularlyperturbedproblemsandtheirapproximations.Forexample,inthe caseofproblemswithaparabolicboundarylayer,theboundary-layerfunction (thatisfiniteinthemaximumnorm)tendstozerointhenormsmentioned aboveas ε → 0[87,33].Inthisbook, maximumnorms areconsistently used.Asarule,weavoidreferencetoworkswhereproblemsforsingularly perturbedordinarydifferentialequationsareconsideredsincesuchresults andtechniquescannot,ingeneral,becarriedovertoproblemsforpartial differentialequations.
Thefirst ε-uniformlyconvergentdifferenceschemesconstructedforsingularlyperturbedproblemsusedtwomainapproaches: fittedoperatormethod and condensingmesh(grid)method/fittedmesh(grid)method.Schemesbased onthefittedoperatormethodwereconstructedin[2]and,independently, constructedandjustifiedin[56](forordinarydifferentialequations);in[15],a schemeforthecondensingmeshmethodwasconstructedandjustified(foran ellipticequation).Forschemesusingcondensingmeshes, ε-uniformconvergenceofthesolutionofadifferenceschemetothesolutionoftheboundary valueproblemisguaranteedbyaspecialchoiceofthedistributionofmesh points(forthegivennumberofnodes).Restrictionsonthechoiceofdifference equationsapproximatingsingularlyperturbedproblems(forensuringthe εuniformconvergenceofthescheme)are,ingeneral,notimposed.Infitted operatormethods, ε-uniformconvergenceofthesolutionofthedifference schemeisachievedbyaspecialchoiceofcoefficientsofthedifferenceequations approximatingthedifferentialproblem.Restrictionsonthedistributionof meshpointsforensuringthe ε-uniformconvergenceoftheschemearenot imposed.
Wementionalsoanapproachrelatedto additivesplittingofasingularity suggestedforsingularlyperturbedproblemsin[12](seealso[11]).Inthis method,basicfunctionsincludespecialfunctionsapproximatingthesingular componentofthesolutionoftheproblem.Inthecaseofsingularlyperturbed
problemsforpartialdifferentialequations,thisapproachwasnotwidelyused becausethesingularcomponentsofsolutionsofboundaryvalueproblemshave theformtoocomplicatedfortheeffectiveconstructionofasystemofbasis functions.Thedifferenceschemesbasedonthemethodofadditivesplitting ofasingularityandconstructedin[12,11]converge ε-uniformlyintheenergy norm.
Afterthepublications[15,56],therewasalargeefforttodevelopfittedoperatormethods.Thefirstbook[26]iscompletelydevotedtothedevelopment ofsuchmethodsforordinarydifferentialequations.Later,fittedoperator methodscontinuedtobeintensivelydeveloped(see,forexample,aseries ofvariantsofthefittedoperatorschemesforellipticequationsin[103]).A comprehensivebibliographyonnumericalmethodsforsingularlyperturbed problemsisgivenin[106].After[15,56],inthecaseofpartialdifferential equations,thefirstfinitedifferenceschemesthatconverge ε-uniformlyinthe maximumnormareconstructedin[29](seealso[30, 1]forthefittedoperator scheme,and[74, 118]fortheschemesoncondensingmeshes).
Notethatfittedoperatormethods(seetheirdescription,e.g.,in[26, 87, 33, 106])haveanadvantageinsimplicitybecausemeshesusedareuniform, andthiscontributedtotheirmorerapidprogresscomparedwithcondensing meshmethods.However,fittedoperatormethodshavearestricteddomainof applicabilityforconstructing ε-uniformlyconvergentnumericalmethods.
Itwasfirstestablishedin[124]thatthereare no ε-uniformlyconvergent schemesbasedonthefittedoperatormethod inthecaseforsingularlyperturbedellipticconvection-diffusionequationsindomainswherepartsofthe boundaryarecharacteristicsofareducedequationand parabolicboundary layers appear.Inthesamepaper,aschemewasconstructedthatconverges ε-uniformly,usingboththefittedoperatormethodfortheapproximationof derivativesalongcharacteristicsofthereducedequationandthecondensing meshmethodfortheapproximationofderivativesinthedirectionorthogonal tothecharacteristics.Theresultingdiscretesolutionsalsomakeitpossible toapproximatethenormalizedderivatives ε-uniformly.
Forparabolicequationswith paraboliclayers itisprovedthatthere existno schemes usingthefittedoperatormethodthatconverge ε-uniformlyinanyof thepapers:
[130]inthecaseofa parabolicboundarylayer and [148]inthecaseofa parabolicinitiallayer
Whenconstructingschemesfor nonlinearproblems,thesituationismuch morecomplicated.In[137](forreaction-diffusionequations)and[139](for convection-diffusionequations),itwasestablishedthatevenforsemilinear ordinarydifferentialequationsthere existnoschemesbasedonthefittedoperatormethod thatconverge ε-uniformly.Similardifficultiesrelatedtothe useoffittedoperatormethodsinnumericalmethodsarediscussedinlater
32] for semilinear ordinary differential equations). Numerical exp eriments that publications (see, e.g., [138, 87, 86] for partial differential equations and [84,
demonstratetheinconsistencyofthefittedoperatormethodforsemilinear
ordinarydifferentialequationareconsideredin[33].Thus,forawideclassof boundaryvalueproblemstherearenoschemesusingfittedoperatormethods thatconverge ε-uniformlyinthemaximumnorm;independentlyofthese, schemesmaybeconstructedusingclassicalfinitedifferenceapproximations orfiniteelementorfinitevolumemethods.
Havingsummarizingtheapproachestotheconstructionof ε-uniformlyconvergentnumericalmethods,wemakeonemoreremark.Inworks[134,140] (seealsodiscussionsin[87, 83]),aclassofproblemsisdistinguishedforellipticandparabolicequationsthatdegenerateontheboundaryofthedomain whosesolutionscontaininitialandparabolicboundarylayers.Itisshown thatforsuchproblems therearenoschemesofthecondensingmeshmethod (forrectangularmeshes)that converge ε-uniformly.Buttheapplicationof bothapproaches—fittedmeshandfittedoperatormethods— makesitpossible toconstructschemesthatconverge ε-uniformly.
Uptonow,practicallyallsingularlyperturbedpartialdifferentialequations, forwhichdifferenceschemesthatconverge ε-uniformlyinthemaximumnorm havebeenconstructed,donotcontainmixedderivatives.Boundaryvalue problemshavebeenconsideredonlyforthesimplestsubdomainsofdimension nothigherthantwo,andtheellipticoperatorinthedifferentialequationsis theLaplaceoperator.Inthecaseofanellipticequationwithmixedderivatives consideredonlyonarectangle,the ε-uniformlyconvergentschemeobtained appearstobetoocomplexandcannotbeextendedtootherdimensionsin geometry[28].Problemsindomainswithcurvilinearboundariesareconsideredonlyinfewpublications.
Itisoneofthegoalsofthepresentbooktoovercomesuchanexisting unsatisfactorystateintheareaofdevelopmentof ε-uniformlyconvergentdifferenceschemes.Anothergoalofthebookis,onmodelproblems,toconsider somemoderntrendsinthedevelopmentofnumericalmethodsforsingularly perturbedproblemsthatrequirefurtherinvestigation.
1.2 Theoretical problemsintheconstructionofdifferenceschemes
Wediscusssome basicprinciples relatedtothefoundationsofthetheory offinitedifferenceschemes,whichariseinthedevelopmentof ε-uniformly convergentnumericalmethods.
Thebehaviourofderivativesofasolutiontosingularlyperturbedboundary valueproblemsmotivatesthetypeofgridandthedifferenceschemesonthat. Thederivativesofthesolutionoftheboundaryvalueprobleminadomainwith asufficientlysmoothboundaryare ε-uniformlyboundedinmostofthedomain andgrowwithoutboundonlyinanarrowsubdomain(theboundarylayer) whosewidthtendstozerowith ε.Theessential anisotropyofthedirectional
derivatives isexhibited.Themaximalvaluesofderivativesofthesolution inaneighborhoodoftheboundarylayerarefoundonlyforthederivative alongthenormaltotheboundaryofthedomain;the k-thnormalderivative intheneighborhoodoftheboundaryisoforder O ε k ,andmoreover,the derivativedecreasesexponentiallytofinitevaluesasthevalue ε k ρ grows, where ρ isthedistancetotheboundary.Thebehaviourofthesolutionofthe boundaryvalueproblemisparticularlycomplicatedinthecaseofdomains withpiecewise-smoothboundaries.
Theobservedanisotropyinthebehaviourofthederivativesofthesolution andthe unboundedgrowthofthederivatives for ε → 0motivatestheuseof essentiallyanisotropicmeshes condensinginaneighborhoodoftheboundary layer.
Notethatthetermsofadifferentialconvection-diffusionequation,involving spatialderivativesacrosstheboundarylayer,becomeunboundedas ε → 0. So,veryclosetotheboundary,inordertoapproximatethesolutionofthe boundaryvalueproblembyasolutionofafinitedifferencescheme,itisnecessarytousemesheswhosestep-sizealongthenormaltotheboundaryisof order o(ε)[138].Underthisrequirementonthemesh-size, errors resultingin theapproximationofthetermsofthedifferentialequation,whenthederivativesintheequationarereplacedbydifferencederivatives,alsogrowwithout bound[138].Thedifferenceschemenolongerapproximatestheboundary valueproblem ε-uniformly.Thisunboundedbehaviouroftermsinthedifferentialequationandtheirdifferenceapproximationsgivesrisetodifficultiesin theproofof ε-uniformconvergenceoftherelevantschemes.
Theviolationofthe monotonicityproperty (see[108])ofaboundaryvalue problemwhengridapproximationsareconstructedevenforthesimplestproblemscanleadtobothlargeerrorsandnonphysicalresults.Forexample,inthe caseofanordinarydifferentialconvection-diffusionequation,whencentered differenceapproximationsonuniformmeshesareused,theerrorinthediscrete solutionoscillatesandgrowswithoutboundiftheparameter ε ismuchless thanthemesh-size[33].
Inthepresenceofmixedderivativesindifferentialequationsandalsointhe caseofproblemsindomainswithcurvilinearboundaries,significantdifficulties ariseintheconstructionofmonotonediscreteapproximationswhen meshes thatcondensegraduallyintheboundarylayer areused(forexample,Bakhvalov’smeshes[15]).Also,fordifferenceschemesthatdonotapproximatethe boundaryvalueproblems ε-uniformly,additionaldifficultiesappearwhenjustifyingthe ε-uniformconvergenceofsolutionsinthemaximumnorm.
Conventionalapproachestoovercomingthespecificproblemsmentioned abovethatariseintheconstructionof ε-uniformlyconvergentnumericalmethodsturnouttobeineffective.
So,inconvection-diffusionproblemstheuseofanartificialviscosityina numericalmethodinordertosuppressoscillationsinthegridsolutionleads toessentialerrorsinaneighborhoodoftheboundarylayer[214,55].
Inthecaseofsingularlyperturbedproblemsthemethodsforthederivation
of aprioriestimates developed forregularproblems (see,e.g.,[67, 68, 69, 37]) resultinestimatesforthederivativesthatdonotdistinguishthedifferent behavioursofderivativesindifferentdirections.Therequirementsimposed onthedataofaregularproblemforobtaining apriori estimatesbythis approachareclosetobeingnecessary.Butestimatesobtainedinthisway forsingularlyperturbedproblemsturnouttobeunabletojustify ε-uniform convergence.
Methodsbasedon asymptoticexpansionsinpowersoftheparameter ε allow onetoconstructapproximationstothesolutionswithanaccuracyuptoa sufficientlyhighorderoftheparameter ε (see,e.g.,[211, 210, 208, 59, 62, 209, 94, 57]andreferencestherein).Butthedetailedbehaviourofthederivativesof thesolutioninvariousdirectionsis,ingeneral,notconsidered.Toconstruct asymptoticexpansionsitisrequiredthatthedataoftheboundaryvalue problempossesssufficientsmoothness.Thisthenrestrictstheclassofboundaryvalueproblemsforwhich ε-uniformlyconvergentnumericalmethodscan bedevelopedandjustified.Thus,theaboveapproachesturnouttobeof littleusefortheconstructionof apriori estimatesrequiredfordeveloping ε-uniformlyconvergentnumericalmethods.
1.3 The mainprinciplesintheconstructionofspecialschemes
Wenowoutlinethe mainprinciples underlyingthedevelopmentofthe εuniformlyconvergentdifferenceschemeswhichareusedinthisbook.Even whennumericalmethodsareconstructedforregularproblemsthathavea complicatedsolution,itisnecessarytousegridapproximationsthatinherit themonotonicitypropertyoftheboundaryvalueproblem[108]inorderto preventtheappearanceofnonphysicaleffectinthesolution.Inthecaseof singularlyperturbedproblemssuchanaturalrequirementleadstocomplicateddifferenceschemes(inthecasewhenmixedderivativesarepresentin theequations,see,e.g.,schemesin[112]forregularproblemsandin[138]for singularlyperturbedboundaryvalueproblems).Fortheclassesofboundary valueproblemsstudiedheretheproblemsthatappeararetofindappropriate classesofgridapproximationsandalsotoobtainsufficientconditions,close tonecessaryones,forthe ε-uniformconvergenceoftheschemesconstructed.
Thestudyofsufficientlywideclassesofboundaryvalueproblemsrequires thedevelopmentofnewapproachestotheconstructionofnumericalmethods and,ingeneral,leadstomorecomplicatedschemes.Primarily,weneed a priori estimatesforthesolutionsthatadequatelyreflectthebehaviourofsuch singularitiesasaboundarylayerinaneighborhoodofdifferentpartsofthe domainboundary.Toconstructspecialschemes,weusethesimpleststandard ε-uniformlymonotoneapproximationstotheboundaryvalueproblem.The
schemesareconstructedonthesimplestpiecewiseuniformmeshesthatensure the ε-uniformconvergenceofthegridsolution.
Thetechniqueforconstructing aprioriestimatesbasedonadecomposition ofthesolution firstconceivedin[118]anddevelopedin[138](seealso[129, 131, 87]andthediscussions,forexample,in[105, 96, 95]foratwo-dimensional problem)wascalledtheShishkindecomposition(see,e.g.,[72, 73]).This techniqueuses thedecompositionofthesolutionintoregularandsingular components and,inthecaseofproblemsindomainswithpiecewisesmooth boundaries,the singularcomponent,inturn,is representedintermsofits componentsasthesumofregularandcornerlayers ofdifferentdimension. Thespecificbehaviouroftheregularcomponent,aswellasofconstituentsof thesingularcomponents,isrevealedusingthe firsttermsoftheirasymptotic expansions withrespecttotheparameter ε.Suchatechniqueallowsusto detectthedistinctionsinthebehaviourofderivativesofthesingularcomponentindifferentdirections,andtoobtainestimatesthatarenecessaryto provethe ε-uniformconvergenceofthedifferenceschemesbeingconstructed. Inthemethodofcondensinggridsconceivedin[128],whichuses piecewise uniformmeshes,thedistributionofmeshpointsinthedirectionacrossthe boundarylayer,normaltotheboundary,isdefinedbyonetransitionparameter,thatis,bythepointatwhichthemeshsizechanges.Intheliterature,such meshesarecalledShishkin’smeshes(see,e.g.,[83]).Fordifferenceschemeson thesemeshes,thedifficultiesinthedevelopmentof ε-uniformlyconvergentgrid methodsareessentiallyreducedthatallowstheconstructionof ε-uniformly monotonedifferenceschemes whichconverge ε-uniformly forrepresentative classesofsingularlyperturbedboundaryvalueproblems.Thefirstnumerical resultsonpiecewiseuniformmesheswereobtainedin[88]foranordinarydifferentialequation,in[89]foraparabolicpartialdifferentialequationwitha parabolicboundarylayer,andin[40]foranellipticconvection-diffusionequationonarectanglewithaparabolicboundarylayerthatisgeneratedonthe characteristicpartsoftheboundary.
An ε-uniformlyconvergentschemeforaprobleminatwo-dimensionaldomainwithacurvilinearboundaryandanequationwithmixedderivatives wasconstructedin1989forthefirsttimein[130]inthecaseofalinear parabolicreaction-diffusionequation.Forsomeboundarylayerproblemsin n-dimensionaldomains,suchsimilarschemeswereconstructedin[131],and in[136,137]forquasilinearequationsinan n-dimensionalslab(seealso[139] (1992)).The ε-uniformlyconvergentschemesinallthesecaseswereconstructedonpiecewiseuniformmeshes.
Theappearanceofpiecewiseuniformmeshes(see,e.g.,[127, 128])that simplifiedtheconstructionandstudyof ε-uniformlyconvergentgridmethods andalsotheprogressintechniquesforobtaining apriori estimates,aswellas theidentificationofalargeclassofproblemsforwhichtheapproachbasedon afittedoperatormethodisrestricted,contributedtothedevelopmentof εuniformlyconvergentdifferenceschemesforpartialdifferentialequations(see someresultsrelatedtothedevelopmentof ε-uniformlyconvergentschemes,
forexample,inthebooks[138, 87, 106, 33],inthesurvey[204],andinthe referencestherein).
1.4 Moderntrendsinthedevelopmentofspecialdifferenceschemes
Inthebookwepointoutsomeproblemsrelatedtonumericalmethodsfor singularlyperturbedproblemsthatrequirefurtherinvestigation.
• Asarule,theorderof ε-uniformconvergenceofdifferenceschemesislow; thereforetheseschemesaretooinefficientforpracticaluse.Thus,the developmentof schemeswithahighconvergenceorder isanimportant problem.
• Apriori estimatesforsolutionsofboundaryvalueproblemslargely“dictatethestructure”ofspecialrobustschemes.Therefore,thedevelopmentofatechniqueforobtaining apriori estimatesfornewclassesof boundaryvalueproblems,inparticular, problemsthathaveseveralperturbationparameters,or problemsforsystemsofequations isanimportanttask.
• Inanumberofboundaryvalueproblems,forexample,forparabolic equationswithmovingboundaryand/orinteriorlayers(see,e.g.,[159]), thecomplicatedsolutionsbringtoocomplicated ε-uniformlyconvergent schemesundertheconstruction.Aproblemofcurrentinterestistofind necessaryandsufficientconditionsfor ε-uniformconvergence ofsuch specialschemes.
• Eveninregularproblems,theconditioningnumberofclassicaldifference schemesgrowswithoutboundwhenthemesh-sizetendstozero.Inthe condensinggridmethod,themesh-sizeinthelayerforafixednumber ofnodescanbearbitrarilysmalldependingontheparameter ε.Thus, thestudyofconditioning,aswellas ε-uniformconditioningofspecial schemes,isanimportantproblem.
Theseproblemsrelatetothefoundationsofthetheoryofnumericalmethods(differenceschemes)forsingularlyperturbedproblemswithcomplicated singularities.
1.5 The contentsofthepresentbook Thecontentsofthebookcanbedividedintotwoparts.PartIis “translation” ofcertainpartsofthebook[138]publishedin1992inRussian,and Part II isa survey ofsomerecentresultsinthedevelopmentofrobustmethods forboundaryvalueproblemswithboundarylayersandothersingularities. Theworkinvolvedinthe “translation” resultedinthedevelopmentindetail ofprinciplesforconstructingmonotonedifferenceschemesforsomeclassesof singularlyperturbedproblems.
InPartI(Chapters2–7)boundaryvalueproblemsareconsideredforelliptic andparabolicreaction-diffusionandconvection-diffusionequations in n-dimensionaldomainswithsmoothandpiecewisesmoothboundaries inthecase whenthe differentialequationscontainmixedderivatives.Hereourgoalis todevelopatechniqueforconstructingandjustifying ε-uniformlyconvergent differenceschemes,basedonclassicaldiscreteapproximations,forboundary valueproblemswithlessrestrictionsontheproblemdata.Inthissituation wearenottooconcernedabouttherateof ε-uniformconvergenceofschemes constructed.Significantattentionisgiventothestudyof conditions thatare necessaryandsufficient forthe ε-uniformconvergence oftherelevantschemes. Whenconstructing specialdifferenceschemesthatconverge ε-uniformly,we usethetechniquesdevelopedfor canonicalproblemsonaslabandaparallelepiped.
PartIIcontainsmaterialpublishedmainlyinthelastfouryears.These areproblemswith boundarylayers,and additionalsingularities generatedby nonsmoothdata,unboundednessofthedomain,andalsobythepresenceof theperturbationvectorparameter.Anotheraspectoftheconsiderationsin thispartisthatwe findboththesolutionanditsderivatives witherrorsthat areindependentoftheperturbationparameters.
In Chapters8 and 13,boundaryvalueproblemsforequationswithperturbationvectorparametersarestudied.InChapter8,problemsareconsidered forascalarparabolicreaction-diffusionequationwiththreescalarparameters, andinChapter13,aproblemisstudiedforasystemoftwoellipticreactiondiffusionequationswithtwoscalarparameters.Intheseproblems,layersof differenttypesarisedependingontherelationbetweenthescalarparameters.
Forgridapproximationsofboundaryvalueproblems withamovingboundarylayer,necessaryandsufficientconditionsfortheschemestoberobust areobtainedin Chapter9.Takingintoaccounttheseconditions,difference schemesareconstructedforaparabolicreaction-diffusionequationinadomainwithmovingboundaries.
Methodsof improvingtheaccuracyofgridsolutions whilepreservingtheir robustnessareexaminedin Chapter10. Conditioning and correctness (or well-posedness)ofadifferenceschemethatisconstructedbasedonaclassical gridapproximationofaboundaryvalueproblemforaparabolicconvection-
diffusionequationonpiecewise-uniformmeshesarediscussedin Chapter12. In Chapter11,differenceschemeson apriori adaptedmeshesaredeveloped.
Promisingapproachestothedevelopmentofrobustmethodsbasedon a posterioriadaptedtechniques arediscussedin Chapter14.Inthecaseof singularlyperturbedproblems,theapproachesknownforregularproblems improvetheconvergenceonlyforfinitevaluesoftheparameter ε,and do notprovidefortherobustnessofthemethod.Here,differenceschemesare discussedthatconverge“almost ε-uniformly”.
InthesurveyChapter14wegivecommentstosomeapproachesintheconstructionofspecialdifferenceschemesandresearchdirections.Inparticular, thefollowingproblemsarediscussed:
• some“applied”problemswhosesolutionsrequirerobustnumerical methods;
• approachestotheconstructionofrobustnumericalmethodsforparabolicequationswithsmoothandpiecewisesmoothboundaryconditions;
• approximationofsolutionsandderivatives;
• specificboundaryvalueproblemsforellipticconvection-diffusionequationssuchasaproblemonanunboundeddomainforanequationwith ascalarparameterandaproblemonarectangleforanequationwitha vectorparameter;
• inthecaseofanunboundeddomain,theconceptofthedomainof essentialdependenceofthesolutionthatallowsrobustschemestobe constructedwhosesolutionsconvergeonboundeddomainsbyincreasing thenumberofnodesinthemeshesused;
• compatibilityconditionsforaproblemonarectanglethatguaranteethe smoothnessofthesolutionanditsregularandsingularcomponentsthat arerequiredfortheconstructionandjustificationofrobustschemes.
1.6 The presentbook Wediscuss theexistingstateoftheart intheareaofthedevelopmentof differenceschemesthatconverge ε-uniformlyinthemaximumnorm.
Thebooks[138,87,33],aswellasthepresentbook,arefocussedonthe developmentofa technologyforconstructingfinitedifferenceschemesthat converge ε-uniformlyinthemaximumnorm forawideclassofsingularlyperturbedboundaryvalueproblems.In[106], ε-uniformlyconvergentnumerical methods areconsideredthatusevarioustechniquesandmethodsdevelopedfor
regularandsingularlyperturbedproblems.Theconvergenceofdiscretemethodsisconsideredin normscorrespondingtotheappliednumericalmethod In[106]thebasicresultsdevelopedin[138]aregiveninacondensedform,and problemsforequationswithmixedderivatives and multi-dimensionalproblems arenotdiscussed
Thebooks[138,87,33]aredevotedtothedevelopmentoffinitedifference schemesforsingularlyperturbedellipticandparabolicequations.Difference schemesareconstructedbasedon standardmonotonefinitedifferenceapproximationsofdifferentialproblemsonpiecewiseuniformmeshes.Thebooks [138,87]emphasizethetheory,and[33]isstrictlynumerical.
Appearingin1988,piecewise-uniformmeshes[127,128]allowustoovercomeanumberofessentialdifficultiesinthedevelopmentandjustification of ε-uniformlyconvergentfinitedifferenceschemesforrepresentativeclasses ofsingularlyperturbedequationswithpartialderivatives.Thedevelopment oftechniquestoderive apriori estimatesonthebasisofthedecomposition methodappliedtothesolutionofboundaryvalueproblems,andalsotechniquestoconstructmonotonediscreteapproximations(ofboundaryvalueproblems)constructedonpiecewiseuniformmeshes,allowthedevelopmentofspecialfinitedifferenceschemesthatareconvergent ε-uniformly.Asaresultof suchaprogress,thebook[138]waswritten.
In[138] ratherwideclassesofellipticandparabolicequations asreactiondiffusion,andconvection-diffusion,havebeenconsideredin n-dimensionaldomainswithsmoothandpiecewisesmoothboundaries,andalsoincomposite domainsinthepresenceofconcentratedsources.Fortheseproblems, techniquestoconstruct ε-uniformlyconvergentschemeshavebeendeveloped,such schemeshavebeenconstructedandtheirconvergencehasbeenstudied.
Thepresentationofresultsderivedin[138]hasbeengiveninaconcentrated formasinareferencebook,sothatitwaspossibletocoverlargeclasses ofboundaryvalueproblems.However,suchaformofexpositionmakesit difficulttousetheminthedevelopmentofnumericalmethodsforequations withpartialderivatives.Itcouldbethatthiscircumstanceisoneofthe reasonsforthedelayinusingresultsfrom[138],aswellasaratherlong delayinthedevelopmentofrobustnumericalmethodsformultidimensional problems.
Thebook [87]isathoroughintroductiontorobustfinitedifferenceschemes. In[87], basicideasandthetechniques thatareusedtoconstructandjustify ε-uniformlyconvergentdifferenceschemeswereconsideredusingmodel problemsasexamples.Here,difference schemes basedon thefittedoperator method and thecondensingmeshmethod havebeenconstructedthatconverge ε-uniformly.Difficultiesthatarisewhenconstructingrobustnumerical methodsforparabolicequationswithparabolicboundarylayershavebeen considered.
Inthebook [33],foranumberofmodelproblemswithboundarylayers, extensive numericalstudies havebeenperformedforrobustdifferenceschemes, andinparticular,forthosethathavebeenconsideredin[138,87].Forthe
quantitativeanalysisofnumericalmethods,anexperimentaltechniquehas beendevelopedwhichallowsustojustify ε-uniformconvergence,andalsoto findtheparametersintheestimatesoftheconvergencerate.A techniquefor experimentalstudy ofspecialdifferenceschemeshasbeendevelopedin[33]. Thistechniquewastestedbynumericalexperimentscarriedoutforlinearand nonlinearproblems,inparticularforellipticconvection-diffusionequations whosesolutionshaveparabolicboundarylayersthatdegenerateonthedomain boundary.Detailsofthe solvers thatareusedinthecomputationshavealso beengiven.
Intheconcludingchaptersofthebook[33],theefficiencyofthetechniques thathavebeendevelopedinthefirsttenchaptersisdemonstratedwhensolvingthe classicalPrandtlproblem forflowofanincompressiblefluidpasta semi-infiniteflatplate[113].ForthenonlinearPrandtlproblem,adifference schemewasconstructedusinganonlinearsolver;inthisproblem, ε = Re 1 isaperturbationparameter,and Re istheReynoldsnumber.Thenumerical methodgives theproblemsolution (thatiscomponentsoftheflowvelocity) and thenormalizedfirst-orderderivatives withanaccuracyclosetoorderone. Computationalexpansesforthesolution (uptoalogarithmicmultiplier,are proportionaltothenumberofmeshpointsused) areindependentoftheparameter ε.
Inthebook [106],numericalmethodshavebeenconsideredforsingularly perturbedboundaryvalueproblemstoordinarydifferentialequationsand alsotoone-dimensionalparabolicandtwo-dimensionalellipticequations(with theLaplaceoperatorasthemainpartofthedifferentialoperator).Various ideasandtechniquesthatareusedinthenumericalanalysisforsingularly perturbeddifferentialequations,andalsoapproachesdevelopedforregular problems,arediscussed. Finitedifferencemethods and finiteelementmethods havealsobeenconsideredforsingularlyperturbedboundaryvalueproblems. Approximationsofsolutionstoboundaryvalueproblemsin variousnorms, inparticularin themaximum and theenergynorms,havebeendiscussed. Significantattentionhasbeendevotedto ε-uniformlyconvergentnumerical methodsinwhich thefittedoperatormethod and thefittedmeshmethod are appliedusing Bakhvalovmeshes and piecewise-uniformmeshes.Aseparate chapterisdevotedto incompressibleNavier-Stokesequations.Thebookcontainsanextensivebibliographyonnumericalmethodsforsingularlyperturbed boundaryvalueproblems.
Thepresentbook isdevotedto asystematicdetaileddevelopmentofapproaches totheconstructionof ε-uniformlyconvergentfinitedifferenceschemes forsomeclassesofsingularlyperturbed boundaryvalueproblemsconsidered in[138].Problemsfor multi-dimensionalellipticandparabolicequationsthat arereaction-diffusionandconvection-diffusion arestudiedinthecasewhen mixedderivativesareinvolvedinthedifferentialequations.Theproblems areconsideredin domainswithpiecewisesmoothandcurvilinearboundaries Boundaryvalueproblemsfrom[138]suchas problemsforequationswithdis-
continuouscoefficientsinthepresenceofconcentratedsources,andalso problemsforconvection-diffusionequationswithcharacteristicpartsontheboundary,arenotconsidered.
The approach presentedhereincludes:
• atechniquetoconstructaprioriestimatesforsolutions ofboundary valueproblemswhichare necessary toconstructandjustifyfinitedifferenceschemes ε-uniformlyconvergentinthemaximumnorm,
• atechniquetoconstructfinitedifferenceschemes basedonstandard discreteapproximationsofboundaryvalueproblems,preservingmonotonicityofdifferentialproblems,
• atechniquetoconstructspecialpiecewiseuniformmeshes(grids) providing ε-uniformconvergenceoftheschemesunderconstruction.
Furthermore,significantattentionisdevotedtotheconsiderationof modern trends inthedevelopmentof ε-uniformlyconvergentnumericalmethodsfor singularlyperturbedproblems.Theinvestigationsofaseriesofmodelproblemscanbeconsideredasrepresentativemodelsofmoregeneralclasses,for whichapproachesfromthefirstpartofthepresentbookcouldbeappliedto thedevelopmentofappropriatenumericalmethods.
Approachesdevelopedinthebook,andresultsderived,shouldpromote furtherdevelopmentofefficientnumericalmethodsfor singularlyperturbed problemswithdifferenttypesofsingularities,andatthesametime,preserve thesamequalitiesalreadywell-establishedinnumericalmethodsforregular boundaryvalueproblems.
Inthepresentbook,aswellinthebooks[138,87,33], ε-uniformlyconvergentschemesareconstructedon simplest piecewiseuniformmeshes,i.e.,mesheshavingonlyone transitionpoint inaneighborhoodoftheboundarylayer. SchemesonBakhvalovmeshes,aswellasonmesheswithfew transitionpoints inaneighborhoodoftheboundarylayer(see,e.g.,[169]),arenotconsidered here.
Thepresentbookandthebooks[138,87,106,33],consideredasonepackage,coverawide spectrumofproblems thatarisewhenconstructingspecial numericalmethodsforproblemswithboundarylayers.Nevertheless,there isalarge gap betweencomplicatedsystemsofnonlinearsingularlyperturbed equationswithpartialderivativeswhich needtobesolved,e.g.,frommathematicalmodelling,andthosesingularlyperturbedproblemsforwhich theoreticallyjustifiednumericalmethodsexist.Wehopethattheresultsgivenhere couldbehelpful inthedevelopmentandanalysisofnewnumericalmethods forproblemsthathaveyettobesolved.
1.7 The audienceforthisbook Thepresentbook,togetherwiththebooks[138,87,106,33],couldbe usefulforscientists-researchers(fromstudentsuptoprofessionals)inthedevelopmentofnumericalmethodsforsingularlyperturbedproblemsandalso forspecialist-researcherswhoareinterestinginmathematicalmodellingand whereproblemswithboundaryandinteriorlayersarisenaturally.
Inthefirstpartofthebookdevotedto multi-dimensionalproblems,techniquesandideasexploitedto deriveaprioriestimatesforthesolutions andto constructthesimplestdiscreteconstructions,i.e.,piecewiseuniformmeshes andstandardmonotonefinitedifferenceoperatorsonthem,arecarefullydiscussedforeachtypeofboundaryvalueproblem.Conditionsimposedon discreteconstructionsarediscussed,anditisshownthatsuch conditionsare closetonecessaryconditions.Techniquesandprinciplesgiveninthebook allowthereadertoconstructindependentlyrobustfinitedifferenceschemes fornewclassesofboundaryvalueproblems,inparticular,forproblemsfrom [138]thatarenottreatedinthepresentbook.
Inthesecondpartofthebook,aconsiderationofmoderntrendsarecarried outonmodelproblems,whichallowtocheckonideasandtechniquesused intheconstruction.Here,wegivereferencestooriginalsourceswheredetails aregivenintheexposedbibliography.Thediscussionofmoderntrendsallows thereadertobemoreconfidentinthisfastgrowingareaofresearch.Thefirst andsecondpartsofthebookcanbestudiedindependently.Thebookcanbe alsohelpfulforresearchersstudyingsingularlyperturbedordinarydifferential equations.
Ifthereaderwishestodevelopnewdifferencemethodsforparticularsingularlyperturbedproblems,itisonlynecessarytoreadPartI.
Ifthereaderwishestoapplydifferencemethodsalreadydevelopedforsingularlyperturbedproblems,thereadershouldpassto PartII.
Chapter 2 Boundary valueproblemsforellipticreaction-diffusion equationsindomainswithsmoothboundaries
Inthischapterboundaryvalueproblemsareconsideredforellipticreactiondiffusionequationsindomainswithsmoothboundaries.Theconstruction offinitedifferenceschemesandthejustificationoftheirconvergenceiscarriedoutusingananalogofthewell-knownsufficientconditionforconvergenceofschemesforregularboundaryvalueproblemsthatisaconsequence ofapproximationbyastablefinitedifferencescheme(itsdescriptionsee,e.g., [79, 108, 100]).Whenappliedtosingularlyperturbedboundaryvalueproblems,this sufficientconditionfor ε-uniformconvergence canbestatedinthe followingway:
ε-uniformconvergenceofafinitedifferenceschemefollowsfrom
ε-uniformapproximationoftheboundaryvalueproblembyan
ε-uniformlystablefinitedifferencescheme.
Ourfinitedifferenceschemesareconstructedbasedonclassicaldifferenceapproximationsofdifferentialequations. ε-uniformapproximationofboundary valueproblemsbyfinitedifferenceschemesisobtainedbyaspecialdistributionofmeshpoints. ε-uniformmonotonicity(and,hence,stability)ofthe finitedifferenceschemesisachievedbymeansofspatialmesheswhosedistributionsofnodesineachcoordinatedirectionareconsistentwitheachother.
2.1 Problem formulation.Theaimoftheresearch Onan n-dimensionaldomain D withsufficientlysmoothboundaryΓ= Γ(D)and D = D Γ, (2.1) where D is, ingeneral,anunboundeddomain,weconsidertheDirichletproblemforelliptic reaction-diffusionequations
Another random document with no related content on Scribd:
The Project Gutenberg eBook of The starstealers This ebook is for the use of anyone anywhere in the United States and most other parts of the world at no cost and with almost no restrictions whatsoever. You may copy it, give it away or re-use it under the terms of the Project Gutenberg License included with this ebook or online at www.gutenberg.org. If you are not located in the United States, you will have to check the laws of the country where you are located before using this eBook.
Title: The star-stealers
Author: Edmond Hamilton
Illustrator: Hugh Rankin
Release date: April 21, 2024 [eBook #73442]
Language: English
Original publication: Indianapolis, IN: Popular Fiction Publishing Company, 1929
Credits: Greg Weeks, Mary Meehan and the Online Distributed Proofreading Team at http://www.pgdp.net
*** START OF THE PROJECT GUTENBERG EBOOK THE STARSTEALERS ***
THE STAR-STEALERS By EDMOND HAMILTON
[Transcriber's Note: This etext was produced from Weird Tales May, February 1929.
Extensive research did not uncover any evidence that the U.S. copyright on this publication was renewed.]
As I stepped into the narrow bridgeroom the pilot at the controls there turned toward me, saluting.
"Alpha Centauri dead ahead, sir," he reported.
"Turn thirty degrees outward," I told him, "and throttle down to eighty light-speeds until we've passed the star."
Instantly the shining levers flicked back under his hands, and as I stepped over to his side I saw the arrows of the speed-dials creeping backward with the slowing of our flight. Then, gazing through the broad windows which formed the room's front side, I watched the interstellar panorama ahead shifting sidewise with the turning of our course.
The narrow bridgeroom lay across the very top of our ship's long, cigar-like hull, and through its windows all the brilliance of the heavens around us lay revealed. Ahead flamed the great double star of Alpha Centauri, two mighty blazing suns which dimmed all else in the heavens, and which crept slowly sidewise as we veered away from them. Toward our right there stretched along the inky skies the far-flung powdered fires of the Galaxy's thronging suns, gemmed with the crimson splendors of Betelgeuse and the clear brilliance of Canopus and the hot white light of Rigel. And straight ahead, now, gleaming out beyond the twin suns we were passing, shone the clear yellow star that was the sun of our own system.
It was the yellow star that I was watching, now, as our ship fled on toward it at eighty times the speed of light; for more than two years had passed since our cruiser had left it, to become a part of that great navy of the Federation of Stars which maintained peace over all the Galaxy. We had gone far with the fleet, in those two years, cruising with it the length and breadth of the Milky Way, patrolling the space-lanes of the Galaxy and helping to crush the occasional pirate ships which appeared to levy toll on the interstellar commerce. And now that an order flashed from the authorities of our own solar system had recalled us home, it was with an unalloyed eagerness
that we looked forward to the moment of our return. The stars we had touched at, the peoples of their worlds, these had been friendly enough toward us, as fellow-members of the great Federation, yet for all their hospitality we had been glad enough to leave them. For though we had long ago become accustomed to the alien and unhuman forms of the different stellar races, from the strange brainmen of Algol to the birdlike people of Sirius, their worlds were not human worlds, not the familiar eight little planets which swung around our own sun, and toward which we were speeding homeward now.
While I mused thus at the window the two circling suns of Alpha Centauri had dropped behind us, and now, with a swift clicking of switches, the pilot beside me turned on our full speed. Within a few minutes our ship was hurtling on at almost a thousand light-speeds, flung forward by the power of our newly invented de-transforming generators, which could produce propulsion-vibrations of almost a thousand times the frequency of the light-vibrations. At this immense velocity, matched by few other craft in the Galaxy, we were leaping through millions of miles of space each second, yet the gleaming yellow star ahead seemed quite unchanged in size.
Abruptly the door behind me clicked open to admit young Dal Nara, the ship's second-officer, descended from a long line of famous interstellar pilots, who grinned at me openly as she saluted.
"Twelve more hours, sir, and we'll be there," she said.
I smiled at her eagerness. "You'll not be sorry to get back to our little sun, will you?" I asked, and she shook her head.
"Not I! It may be just a pin-head beside Canopus and the rest, but there's no place like it in the Galaxy. I'm wondering, though, what made them call us back to the fleet so suddenly."
My own face clouded, at that. "I don't know," I said, slowly. "It's almost unprecedented for any star to call one of its ships back from the Federation fleet, but there must have been some reason——"
"Well," she said cheerfully, turning toward the door, "it doesn't matter what the reason is, so long as it means a trip home. The crew is worse than I am—they're scrapping the generators down in the engine-room to get another light-speed out of them."
I laughed as the door clicked shut behind her, but as I turned back to the window the question she had voiced rose again in my mind, and I gazed thoughtfully toward the yellow star ahead. For as I had told Dal Nara, it was a well-nigh unheard-of thing for any star to recall one of its cruisers from the great fleet of the Federation. Including as it did every peopled star in the Galaxy, the Federation relied entirely upon the fleet to police the interstellar spaces, and to that fleet each star contributed its quota of cruisers. Only a last extremity, I knew, would ever induce any star to recall one of its ships, yet the message flashed to our ship had ordered us to return to the solar system at full speed and report at the Bureau of Astronomical Knowledge, on Neptune. Whatever was behind the order, I thought, I would learn soon enough, for we were now speeding over the last lap of our homeward journey; so I strove to put the matter from my mind for the time being.
With an odd persistence, though, the question continued to trouble my thoughts in the hours that followed, and when we finally swept in toward the solar system twelve hours later, it was with a certain abstractedness that I watched the slow largening of the yellow star that was our sun. Our velocity had slackened steadily as we approached that star, and we were moving at a bare one light-speed when we finally swept down toward its outermost, far-swinging planet, Neptune, the solar system's point of arrival and departure for all interstellar commerce. Even this speed we reduced still further as we sped past Neptune's single circling moon and down through the crowded shipping-lanes toward the surface of the planet itself.
Fifty miles above its surface all sight of the planet beneath was shut off by the thousands of great ships which hung in dense masses above it—that vast tangle of interstellar traffic which makes the great planet the terror of all inexperienced pilots. From horizon to
horizon, it seemed, the ships crowded upon each other, drawn from every quarter of the Galaxy. Huge grain-boats from Betelgeuse, vast, palatial liners from Arcturus and Vega, ship-loads of radium ores from the worlds that circle giant Antares, long, swift mail-boats from distant Deneb—all these and myriad others swirled and circled in one great mass above the planet, dropping down one by one as the official traffic-directors flashed from their own boats the brilliant signals which allowed a lucky one to descend. And through occasional rifts in the crowded mass of ships could be glimpsed the interplanetary traffic of the lower levels, a swarm of swift little boats which darted ceaselessly back and forth on their comparatively short journeys, ferrying crowds of passengers to Jupiter and Venus and Earth, seeming like little toy-boats beside the mighty bulks of the great interstellar ships above them.
As our own cruiser drove down toward the mass of traffic, though, it cleared away from before us instantly; for the symbol of the Federation on our bows was known from Canopus to Fomalhaut, and the cruisers of its fleet were respected by all the traffic of the Galaxy. Arrowing down through this suddenly opened lane we sped smoothly down toward the planet's surface, hovering for a moment above its perplexing maze of white buildings and green gardens, and then slanting down toward the mighty flat-roofed building which housed the Bureau of Astronomical Knowledge. As we sped down toward its roof I could not but contrast the warm, sunny green panorama beneath with the icy desert which the planet had been until two hundred thousand years before, when the scientists of the solar system had devised the great heat-transmitters which catch the sun's heat near its blazing surface and fling it out as high-frequency vibrations to the receiving-apparatus on Neptune, to be transformed back into the heat which warms this world. In a moment, though, we were landing gently upon the broad roof, upon which rested scores of other shining cruisers whose crews stood outside them watching our arrival.
Five minutes later I was whirling downward through the building's interior in one of the automatic little cone-elevators, out of which I
