Front Tracking for Hyperbolic Conservation Laws (Applied Mathematical Sciences, 152)

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. From the reviews: "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. Cover 1 Applied Mathematical Sciences 152 2 Front Tracking for Hyperbolic Conservation Laws 4 ISBN 9783540432890, 9783642239106 5 Preface 8 Contents 12 1 Introduction 14 1.1 Notes 31 Exercises 32 2 Scalar Conservation Laws 36 2.1 Entropy Conditions 37 2.2 The Riemann Problem 43 2.3 Front Tracking 49 2.4 Existence and Uniqueness 57 2.5 Notes 69 Exercises 71 3 A Short Course in Difference Methods 76 3.1 Conservative Methods 76 3.2 Error Estimates 94 3.3 APriori Error Estimates 105 3.4 Measure-Valued Solutions 112 3.5 Notes 125 Exercises 125 4 Multidimensional Scalar Conservation Laws 130 4.1 Dimensional Splitting Methods 130 4.2 Dimensional Splitting and Front Tracking 139 4.3 Convergence Rates 147 4.4 Operator Splitting: Diffusion 159 4.5 Operator Splitting: Source 166 4.6 Notes 170 Exercises 171 5 The Riemann Problem for Systems 176 5.1 Hyperbolicity and Some Examples 177 5.2 Rarefaction Waves 181 5.3 The Hugoniot Locus: The Shock Curves 187 5.4 The Entropy Condition 193 5.5 The Solution of the Riemann Problem 201 5.6 Notes 213 Exercises 214 6 Existence of Solutions of the Cauchy Problem for Systems 218 6.1 Front Tracking for Systems 219 6.2 Convergence 231 6.3 Notes 242 Exercises 243 7 Well-Posedness of the Cauchy Problem for Systems 246 7.1 Stability 251 7.2 Uniqueness 278 7.3 Notes 298 Exercises 299 Appendix A Total Variation, Compactness, etc. 300 A.1 Notes 311 Exercises 311 Appendix B The Method of Vanishing Viscosity 312 B.1 Notes 325 Exercises 325 Appendix C Answers and Hints 328 References 362 Index 372 3642239102,9783642239106,3540432892,9783540432890 Springer 2011 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. From the reviews:"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 Front Matter....Pages i-xii Introduction....Pages 1-21 Scalar Conservation Laws....Pages 23-62 A Short Course in Difference Methods....Pages 63-116 Multidimensional Scalar Conservation Laws....Pages 117-162 The Riemann Problem for Systems....Pages 163-204 Existence of Solutions of the Cauchy Problem for Systems....Pages 205-231 Well-Posedness of the Cauchy Problem for Systems....Pages 233-286 Back Matter....Pages 287-361 1. Introduction -- 2. Scalar Conservation Laws -- 3. A Short Course In Difference Methods -- 4. Multidimensional Scalar Conservation Laws -- 5. The Riemann Problem For Systems -- 6. Existence Of Solutions Of The Cauchy Problem -- 7. Well-posedness Of The Cauchy Problem. Helge Holden, Nils Henrik Risebro. Includes Bibliographical References (p. [351]-359) And Index. This book offers a detailed, rigorous, and self-contained presentation of the theory of hyperbolic conservation laws from the basic theory to the forefront of research. It contains details and information about numerical approximation for the Cauchy problem.