Front Tracking for Hyperbolic Conservation Laws
Seiten
2002
|
1., st ed 2002. Corr. 2nd printing
Springer Berlin (Verlag)
978-3-540-43289-0 (ISBN)
Springer Berlin (Verlag)
978-3-540-43289-0 (ISBN)
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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. The text offers extensive examples, exercises with hints and answers and comprehensive appendices.
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.
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.
1 Introducation * 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 Wellposedness of the Cauchy Problem * A Total Variation, Compactedness, etc. * B The Method of Vanishing Viscosity * C Answers and Hints * References * Index
| Reihe/Serie | Applied Mathematical Sciences ; 152 |
|---|---|
| Sprache | englisch |
| Maße | 155 x 235 mm |
| Gewicht | 658 g |
| Einbandart | gebunden |
| Themenwelt | Mathematik / Informatik ► Mathematik |
| Schlagworte | conservation laws • front tracking • hyperbolic partial differential equations • Hyperbolisches System |
| ISBN-10 | 3-540-43289-2 / 3540432892 |
| ISBN-13 | 978-3-540-43289-0 / 9783540432890 |
| Zustand | Neuware |
| Informationen gemäß Produktsicherheitsverordnung (GPSR) | |
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