# Front Tracking for Hyperbolic Conservation Laws

Springer Science & Business Media, 15.5.2007 - 264 sivua
Hyperbolic conservation laws are central in the theory of nonlinear partial differential equations, and in many applications in science and technology. In this book the reader is given a detailed, rigorous, and self-contained presentation of the theory of hyperbolic conservation laws from the basic theory up to the research front. The approach is constructive, and the mathematical approach using front tracking can be applied directly as a numerical method. After a short introduction on the fundamental properties of conservation laws, the theory of scalar conservation laws in one dimension is treated in detail, showing the stability of the Cauchy problem using front tracking. The extension to multidimensional scalar conservation laws is obtained using dimensional splitting. Inhomogeneous equations and equations with diffusive terms are included as well as a discussion of convergence rates. The classical theory of Kruzkov and Kuznetsov is covered. Systems of conservation laws in one dimension are treated in detail, starting with the solution of the Riemann problem. Solutions of the Cauchy problem are proved to exist in a constructive manner using front tracking, amenable to numerical computations. The book includes a detailed discussion of the very recent proof of wellposedness of the Cauchy problem for one-dimensional hyperbolic conservation laws. The book includes a chapter on traditional finite difference methods for hyperbolic conservation laws with error estimates and a section on measure valued solutions. Extensive examples are given, and many exercises are included with hints and answers. Additional background material not easily available elsewhere is given in appendices.

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### Sisältö

 Introduction 1 11 Notes 18 Scalar Conservation Laws 23 21 Entropy Conditions 24 22 The Riemann Problem 30 23 Front Tracking 36 24 Existence and Uniqueness 44 25 Notes 56
 52 Rarefaction Waves 170 The Shock Curves 176 54 The Entropy Condition 182 55 The Solution of the Riemann Problem 190 56 Notes 202 Existence of Solutions of the Cauchy Problem for Systems 207 61 Front Tracking for Systems 208 62 Convergence 220

 A Short Course in Difference Methods 63 32 Error Estimates 81 33 A Priori Error Estimates 92 34 Measure Valued Solutions 99 35 Notes 112 Multidimensional Scalar Conservation Laws 117 42 Dimensional Splitting and Front Tracking 127 43 Convergence Rates 134 Diffusion 147 Source 154 46 Notes 158 The Riemann Problem for Systems 165 51 Hyperbolicity and Some Examples 166
 63 Notes 231 WellPosedness of the Cauchy Problem for Systems 235 71 Stability 240 72 Uniqueness 267 73 Notes 287 Total Variation Compactness etc 289 A1 Notes 300 The Method of Vanishing Viscosity 301 B1 Notes 314 Answers and Hints 317 References 351 Index 361 Tekijänoikeudet

### Suositut otteet

Sivu 351 - D. Amadori and RM Colombo. Viscosity solutions and Standard Riemann Semigroup for conservation laws with boundary. Rend. Sem. Mat. Univ. Padova 99 (l998).

### Viitteet tähän teokseen

 Control and Boundary AnalysisRajoitettu esikatselu - 2005
 Control Methods in PDE-dynamical Systems: AMS-IMS-SIAM Joint Summer Research ...Fabio AnconaEsikatselu ei käytettävissä - 2007
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