Front Tracking for Hyperbolic Conservation LawsSpringer Science & Business Media, 15 de mai. de 2007 - 264 páginas 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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Conteúdo
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 |
351 | |
361 | |
Outras edições - Ver todos
Front Tracking for Hyperbolic Conservation Laws Helge Holden,Nils H. Risebro Visualização parcial - 2013 |
Front Tracking for Hyperbolic Conservation Laws Helge Holden,Nils H. Risebro Visualização parcial - 2007 |
Front Tracking for Hyperbolic Conservation Laws Helge Holden,Nils Henrik Risebro Visualização parcial - 2011 |
Termos e frases comuns
approximation Assume bounded bounded variation collision compute conservation laws consider continuous function converges convex define denote dimensional splitting discontinuity dt dx dx dt dx dy dy ds eigenvalues eigenvectors entropy solution estimate finite number flux function front tracking Furthermore genuinely nonlinear Helly's theorem Hence implies inequality initial data initial value problem integral interaction interval Kružkov entropy condition Lemma linearly degenerate Lipschitz continuous Math method monotone notation obtain Partial Differential Equations piecewise constant piecewise linear proof prove rarefaction curves rarefaction wave Riemann problem satisfies scalar conservation laws scheme sequence shallow-water equations shock solve speed systems of conservation T.V. uo t₁ test function Theorem Theory tn+1 total variation totally bounded u₁ unique uo(x W₂ weak solution zero მე Ալ
Passagens mais conhecidas
Página 351 - D. Amadori and RM Colombo. Viscosity solutions and Standard Riemann Semigroup for conservation laws with boundary. Rend. Sem. Mat. Univ. Padova 99 (l998).