Front Tracking for Hyperbolic Conservation Laws PDF
Preview Front Tracking for Hyperbolic Conservation Laws
Applied Mathematical Sciences Volume 152 Editors S.S. Antman J.E. Marsden L. Sirovich Advisors J.K. Hale P. Holmes J. Keener J. Keller B.J. Matkowsky A. Mielke C.S. Peskin K.R. Sreenivasan Forfurthervolumes: http://www.springer.com/series/34 Helge Holden • Nils Henrik Risebro Front Tracking for Hyperbolic Conservation Laws Helge Holden Nils Henrik Risebro DepartmentofMathematics DepartmentofMathematics Norwegian University of Science University of Oslo and Technology Blindern Trondheim Norway Norway [email protected] [email protected] Editors: S.S.Antman J.E.Marsden L.Sirovich DepartmentofMathematics (1942–2010) LaboratoryofApplied and CaliforniaInstitute Mathematics InstituteforPhysicalScience ofTechnology Departmentof andTechnology USA BiomathematicalSciences UniversityofMaryland Mount Sinai School of Medicine CollegePark NewYork,NY10029-6574 MD20742-4015 USA USA [email protected] [email protected] ISSN0066-5452 ISBN978-3-540-43289-0 (hardcover) e-ISBN978-3-642-23911-3 ISBN978-3-642-23910-6 (softcover) DOI10.1007/978-3-642-23911-3 Springer Heidelberg Dordrecht LondonNew York Library of Congress Control Number: 2011939410 MathematicsSubjectClassification(2000):35Lxx, 35L65, 58J45 © Springer-Verlag Berlin Heidelberg 2002, Corrected printing 2007, First softcover printing 2011 This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilm or in any other way, and storage in data banks. Duplication of this publication or parts thereof is permitted only under the provisions of the German Copyright Law of September 9, 1965, in its current version, and permission for use must always be obtained from Springer. Violations are liable to prosecution under the German Copyright Law. The use of general descriptive names, registered names, trademarks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. Printedonacid-freepaper Springer is part of Springer Science+Business Media (www.springer.com) In memory of Raphael, who started it all Preface Vse sqastlivye sem(cid:2)i poho(cid:3)i drug na druga, ka(cid:3)da(cid:4) nesqastliva(cid:4) sem(cid:2)(cid:4) nesqastliva po-svoemu.1 Lev Tolsto(cid:5), Anna Karenina (1875) While it is not strictly speaking true that all linear partial differential equations are the same, the theory that encompasses these equations can be considered well developed (and these are the happy families). Large classes of linear partial differential equations can be studied using linear functional analysis, which was developed in part as a tool to investigate important linear differential equations. In contrast to the well-understood (and well-studied) classes of linear partial differential equations, each nonlinear equation presents its own particular difficulties. Nevertheless, over the last forty years some rather general classes of nonlinear partial differential equations have been stud- ied and at least partly understood. These include the theory of viscosity solutions for Hamilton–Jacobi equations, the theory of Korteweg–de Vries equations, as well as the theory of hyperbolic conservation laws. The purpose of this book is to present the modern theory of hyperbolic conservation laws in a largely self-contained manner. In contrast to the modern theory of linear partial differential equations, the mathematician 1Allhappyfamiliesresembleoneanother,buteachunhappyfamilyisunhappyinits ownway(LeoTolstoy,Anna Karenina). vii viii Preface interestedinnonlinearhyperbolicconservationlawsdoesnothavetocover a large body of general theory to understand the results. Therefore, to follow the presentation in this book (with some minor exceptions), the readerdoesnothavetobefamiliarwithmanycomplicatedfunctionspaces, nor does he or she have to know much theory of linear partial differential equations. The methods used in this book are almost exclusively constructive, and largelybasedonthefront-trackingconstruction.Wefeelthatthisgivesthe reader an intuitive feeling for the nonlinear phenomena that are described by conservation laws. In addition, front tracking is a viable numerical tool, and our book is also suitable for practical scientists interested in computations. We focus on scalar conservation laws in several space dimensions and systems of hyperbolic conservation laws in one space dimension. In the scalar case we first discuss the one-dimensional case before we consider its multidimensional generalization. Multidimensional systems will not be treated.Formultidimensionalequationswecombinefronttrackingwiththe method of dimensional splitting. We have included a chapter on standard difference methods that provides a brief introduction to the fundamentals of difference methods for conservation laws. This book has grown out of courses we have given over some years: full-semester courses at the Norwegian University of Science and Technol- ogy and the University of Oslo, as well as shorter courses at Universit¨a¨t Kaiserslautern and S.I.S.S.A., Trieste. We have taught this material for graduate and advanced undergraduate students. A solid background in real analysis and integration theory is an advantage, but key results concerning compactness and functions of bounded variation are proved in Appendix A. Our main audience consists of students and researchers interested in analytical properties as well as numerical techniques for hyperbolic conservation laws. We havebenefited from the kind adviceand careful proofreadingof var- ious versions of this manuscript by several friends and colleagues, among themPetterI.Gustafson,RunarHoldahl,HelgeKristianJenssen,Kenneth H. Karlsen, Odd Kolbjørnsen, Kjetil Magnus Larsen, Knut-Andreas Lie, Achim Schroll. Special thanks are due to Harald Hanche-Olsen, who has helped us on several occasions with both mathematical and TTTEEEX-nical is- sues.WearealsogratefultoTrondIden,fromOrdkommisjonen,forhelping with technical issues and software for making the figures. Ourresearchhasbeen supportedinpartbythe BeMatAprogramofthe Research Council of Norway. A list of corrections can be found at www.math.ntnu.no/~holden/FrontBook/ Whenever you find an error, please send us an email about it. Preface ix Thelogicalinterdependenceofthematerialinthisbookisdepictedinthe diagrambelow.Themainline,Chapters1,2, 5–7,hasmostoftheemphasis onthetheoryforsystemsofconservationlawsinonespacedimension.An- other possible track is Chapters 1–4, with emphasis on numerical methods and theory for scalar equations in one and several space dimensions. Chapter1 Chapter2 Sections3.1–2 Sections3.3–4 Chapter5 Chapter 4 Chapter6 Chapter7 Inthecorrectedprintingofthefirsteditionwehaveusedtheopportunity to correctmany smallererrors.We aregratefulto thosewho havegivenus feedback, in particular, F. Gossler, X. Raynaud, M. Rejske, and O. Sete.
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