Overview

"Hyperbolic conservation laws are central in the theory of nonlinear partial differential equations and in science and technology. The reader is given a self-contained presentation using front tracking, which is also a numerical method. The multidimensional scalar case and the case of systems on the line are treated in detail. A chapter on finite differences is included. ""It is already one of the few best digests on this topic. The present book is an excellent compromise between theory and practice. Students will appreciate the lively and accurate style."" D. Serre, MathSciNet ""I have read the book with great pleasure, and I can recommend it to experts as well as students. It can also be used for reliable and very exciting basis for a one-semester graduate course."" S. Noelle, Book review, German Math. Soc. ""Making it an ideal first book for the theory of nonlinear partial differential equations...an excellent reference for a graduate course on nonlinear conservation laws."" M. Laforest, Comp. Phys. Comm."

Full Product Details

Author:   Helge Holden ,  Nils H. Risebro
Publisher:   Springer-Verlag Berlin and Heidelberg GmbH & Co. KG
Imprint:   Springer-Verlag Berlin and Heidelberg GmbH & Co. K
Edition:   Softcover reprint of the original 1st ed. 2002
Volume:   152
Dimensions:   Width: 15.50cm , Height: 2.00cm , Length: 23.50cm
Weight:   0.581kg
ISBN:  

9783642627972

ISBN 10:   3642627978
Pages:   264
Publication Date:   23 August 2014
Audience:   Professional and scholarly ,  Professional & Vocational
Format:   Paperback
Publisher's Status:   Active
Availability:   Manufactured on demand  
We will order this item for you from a manufactured on demand supplier.

Table of Contents

1 Introduction.- 1.1 Notes.- 2 Scalar Conservation Laws.- 2.1 Entropy Conditions.- 2.2 The Riemann Problem.- 2.3 Front Tracking.- 2.4 Existence and Uniqueness.- 2.5 Notes.- 3 A Short Course in Difference Methods.- 3.1 ConservativeMethods.- 3.2 Error Estimates.- 3.3 APriori Error Estimates.- 3.4 Measure-Valued Solutions.- 3.5 Notes.- 4 Multidimensional Scalar Conservation Laws.- 4.1 Dimensional SplittingMethods.- 4.2 Dimensional Splitting and Front Tracking.- 4.3 Convergence Rates.- 4.4 Operator Splitting: Diffusion.- 4.5 Operator Splitting: Source.- 4.6 Notes.- 5 The Riemann Problem for Systems.- 5.1 Hyperbolicity and Some Examples.- 5.2 Rarefaction Waves.- 5.3 The Hugoniot Locus: The Shock Curves.- 5.4 The Entropy Condition.- 5.5 The Solution of the Riemann Problem.- 5.6 Notes.- 6 Existence of Solutions of the Cauchy Problem.- 6.1 Front Tracking for Systems.- 6.2 Convergence.- 6.3 Notes.- 7 Well-Posedness of the Cauchy Problem.- 7.1 Stability.- 7.2 Uniqueness.- 7.3 Notes.- A Total Variation, Compactness, etc..- A.1 Notes.- B The Method of Vanishing Viscosity.- B.1 Notes.- C Answers and Hints.- References.

Reviews

"From the reviews: ""The book under review provides a self-contained, thorough, and modern account of the mathematical theory of hyperbolic conservation laws. … gives a detailed treatment of the existence, uniqueness, and stability of solutions to a single conservation law in several space dimensions and to systems in one dimension. This book … is a timely contribution since it summarizes recent and efficient solutions to the question of well-posedness. This book would serve as an excellent reference for a graduate course on nonlinear conservation laws … ."" (M. Laforest, Computer Physics Communications, Vol. 155, 2003) ""The present book is an excellent compromise between theory and practice. Since it contains a lot of theorems, with full proofs, it is a true piece of mathematical analysis. On the other hand, it displays a lot of details and information about numerical approximation for the Cauchy problem. Thus it will be of interest for a wide audience. Students will appreciate the lively and accurate style … . this text is suitable for graduate courses in PDEs and numerical analysis."" (Denis Serre, Mathematical Reviews, 2003 e)"

From the reviews: MATHEMATICAL REVIEWS ...distinguished in the sense that, although its main scope is front tracking...it addresses a larger audience, being also concerned with numerics and applications...The present book is an excellent compromise between theory and practice. Since it contains a lot of theorems, with full proofs, it is a true piece of mathematical analysis. On the other hand, it displays a lot of details and information about numerical approximation for the Cauchy problem. Thus it will be of interest for a wide audience. Students will appreciate the lively and accurate style, the numerous exercises (55 in all) and the fact that the authors systematically avoid side or exotic topics. As mentioned on the back cover, this text is suitable for graduate courses in PDEs and numerical analysis. Since most advanced analytical material is given in appendices, it does not require much background. The book under review provides a self-contained, thorough, and modern account of the mathematical theory of hyperbolic conservation laws. ... gives a detailed treatment of the existence, uniqueness, and stability of solutions to a single conservation law in several space dimensions and to systems in one dimension. This book ... is a timely contribution since it summarizes recent and efficient solutions to the question of well-posedness. This book would serve as an excellent reference for a graduate course on nonlinear conservation laws ... . (M. Laforest, Computer Physics Communications, Vol. 155, 2003) The present book is an excellent compromise between theory and practice. Since it contains a lot of theorems, with full proofs, it is a true piece of mathematical analysis. On the other hand, it displays a lot of details and information about numerical approximation for the Cauchy problem. Thus it will be of interest for a wide audience. Students will appreciate the lively and accurate style ... . this text is suitable for graduate courses in PDEs and numerical analysis. (Denis Serre, Mathematical Reviews, 2003 e)

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