Front Tracking for Hyperbolic Conservation LawsSpringer, 27 de nov. de 2013 - 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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... sequence such that φ(m)k → φ(m) for all m, then 〈h, φk 〉→〈h, φ〉. Hence h is a distribution. Furthermore, if h is a distribution on R, it possesses distributional derivatives of any order, for we define 〈 h(m),φ 〉 := (−1)m 〈 h ...
... sequence such that φ(m)k → φ(m) for all m, then 〈h, φk 〉→〈h, φ〉. Hence h is a distribution. Furthermore, if h is a distribution on R, it possesses distributional derivatives of any order, for we define 〈 h(m),φ 〉 := (−1)m 〈 h ...
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... finding u by solving a sequence of linearized equations. Fix n e N. For t in the interval (m/n, (m + 1)/n and m > 0, let v, solve (wn): + Un (a, m/n) (wn), = 0, and set vn (a,0) = u0(a). Then show that 7", 20 1. Introduction.
... finding u by solving a sequence of linearized equations. Fix n e N. For t in the interval (m/n, (m + 1)/n and m > 0, let v, solve (wn): + Un (a, m/n) (wn), = 0, and set vn (a,0) = u0(a). Then show that 7", 20 1. Introduction.
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Conteúdo
1 | |
22 | |
A Short Course in Difference Methods | 63 |
Multidimensional Scalar Conservation Laws | 117 |
The Riemann Problem for Systems | 163 |
Existence of Solutions of the Cauchy Problem | 205 |
WellPosedness of the Cauchy Problem | 233 |
A Total Variation Compactness etc | 287 |
B The Method of Vanishing Viscosity | 299 |
Answers and Hints 315 | 314 |
References | 349 |
Index 359 | 358 |
Outras edições - Ver todos
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 |
Front Tracking for Hyperbolic Conservation Laws Helge Holden,Nils Henrik Risebro Visualização parcial - 2015 |
Termos e frases comuns
bounded variation breakpoints Bressan Cauchy problem collision compact compute conclude conservation laws consider continuous function converges convex define Differential Equations dimensional splitting dy ds eigenvalues eigenvectors entropy solution estimate exists f|Lip finite number flux function front tracking Furthermore genuinely nonlinear given heat equation Hence hyperbolic conservation laws implicit function theorem implies inequality initial data initial value problem integral interaction interval Kružkov entropy condition Lemma limit linearly degenerate Lipschitz continuous measure-valued solution method monotone nonnegative test function notation Ó Example obtain one-dimensional p+q+1 piecewise linear proof prove Rankine–Hugoniot condition rarefaction curves rarefaction wave result Riemann problem right-hand side satisfies scalar conservation law scheme sequence shallow-water equations shock solution of 3.1 speed test function theory total variation totally bounded un(x wave curves wave family weak solution Young measure