Front Tracking for Hyperbolic Conservation Laws

374 Pages·2011·6.453 MB·English

by  Helge Holden, Nils H. Risebro (auth.)

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.