Front Tracking for Hyperbolic Conservation Laws

Etukansi
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
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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).

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