Shock Waves In Conservation Laws With Physical Viscosity
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Shock Waves In Conservation Laws With Physical Viscosity
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Author : Tai-Ping Liu
language : en
Publisher:
Release Date : 2014
Shock Waves In Conservation Laws With Physical Viscosity written by Tai-Ping Liu and has been published by this book supported file pdf, txt, epub, kindle and other format this book has been release on 2014 with Conservation laws (Mathematics) categories.
We study the perturbation of a shock wave in conservation laws with physical viscosity. We obtain the detailed pointwise estimates of the solutions. In particular, we show that the solution converges to a translated shock profile. The strength of the perturbation and that of the shock are assumed to be small, but independent. Our assumptions on the viscosity matrix are general so that our results apply to the Navier-Stokes equations for the compressible fluid and the full system of magnetohydrodynamics, including the cases of multiple eigenvalues in the transversal fields, as long as the shock is classical. Our analysis depends on accurate construction of an approximate Green's function. The form of the ansatz for the perturbation is carefully constructed and is sufficiently tight so that we can close the nonlinear term through the Duhamel's principle.
Shock Waves In Conservation Laws With Physical Viscosity
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Author : Tai-Ping Liu
language : en
Publisher: American Mathematical Soc.
Release Date : 2015-02-06
Shock Waves In Conservation Laws With Physical Viscosity written by Tai-Ping Liu and has been published by American Mathematical Soc. this book supported file pdf, txt, epub, kindle and other format this book has been release on 2015-02-06 with Mathematics categories.
The authors study the perturbation of a shock wave in conservation laws with physical viscosity. They obtain the detailed pointwise estimates of the solutions. In particular, they show that the solution converges to a translated shock profile. The strength of the perturbation and that of the shock are assumed to be small but independent. The authors' assumptions on the viscosity matrix are general so that their results apply to the Navier-Stokes equations for the compressible fluid and the full system of magnetohydrodynamics, including the cases of multiple eigenvalues in the transversal fields, as long as the shock is classical. The authors' analysis depends on accurate construction of an approximate Green's function. The form of the ansatz for the perturbation is carefully constructed and is sufficiently tight so that the author can close the nonlinear term through Duhamel's principle.
Shock Waves
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Author : Tai-Ping Liu
language : en
Publisher: American Mathematical Soc.
Release Date : 2021-10-12
Shock Waves written by Tai-Ping Liu and has been published by American Mathematical Soc. this book supported file pdf, txt, epub, kindle and other format this book has been release on 2021-10-12 with Education categories.
This book presents the fundamentals of the shock wave theory. The first part of the book, Chapters 1 through 5, covers the basic elements of the shock wave theory by analyzing the scalar conservation laws. The main focus of the analysis is on the explicit solution behavior. This first part of the book requires only a course in multi-variable calculus, and can be used as a text for an undergraduate topics course. In the second part of the book, Chapters 6 through 9, this general theory is used to study systems of hyperbolic conservation laws. This is a most significant well-posedness theory for weak solutions of quasilinear evolutionary partial differential equations. The final part of the book, Chapters 10 through 14, returns to the original subject of the shock wave theory by focusing on specific physical models. Potentially interesting questions and research directions are also raised in these chapters. The book can serve as an introductory text for advanced undergraduate students and for graduate students in mathematics, engineering, and physical sciences. Each chapter ends with suggestions for further reading and exercises for students.
Hyperbolic And Viscous Conservation Laws
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Author : Tai-Ping Liu
language : en
Publisher: SIAM
Release Date : 2000-01-01
Hyperbolic And Viscous Conservation Laws written by Tai-Ping Liu and has been published by SIAM this book supported file pdf, txt, epub, kindle and other format this book has been release on 2000-01-01 with Mathematics categories.
Here is an in-depth, up-to-date analysis of wave interactions for general systems of hyperbolic and viscous conservation laws. This self-contained study of shock waves explains the new wave phenomena from both a physical and a mathematical standpoint. The analysis is useful for the study of various physical situations, including nonlinear elasticity, magnetohydrodynamics, multiphase flows, combustion, and classical gas dynamics shocks. The central issue throughout the book is the understanding of nonlinear wave interactions.
Hyperbolic And Viscous Conservation Laws
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Author : Tai-Ping Liu
language : en
Publisher: SIAM
Release Date : 2000-01-01
Hyperbolic And Viscous Conservation Laws written by Tai-Ping Liu and has been published by SIAM this book supported file pdf, txt, epub, kindle and other format this book has been release on 2000-01-01 with Mathematics categories.
An in-depth analysis of wave interactions for general systems of hyperbolic and viscous conservation laws.
Viscous Profiles And Numerical Methods For Shock Waves
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Author : Michael Shearer
language : en
Publisher: SIAM
Release Date : 1991-01-01
Viscous Profiles And Numerical Methods For Shock Waves written by Michael Shearer and has been published by SIAM this book supported file pdf, txt, epub, kindle and other format this book has been release on 1991-01-01 with Science categories.
One strongly represented theme is the power of ideas from dynamical systems that are being adapted and developed in the context of shock waves.
Quasilinear Hyperbolic Systems And Dissipative Mechanisms
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Author : Ling Hsiao
language : en
Publisher: World Scientific
Release Date : 1998-02-24
Quasilinear Hyperbolic Systems And Dissipative Mechanisms written by Ling Hsiao and has been published by World Scientific this book supported file pdf, txt, epub, kindle and other format this book has been release on 1998-02-24 with Mathematics categories.
This book introduces the recent developments in the subject of quasilinear hyperbolic systems with dissipation, such as frictional damping, relaxation, viscosity and heat diffusion. The mathematical theory behind this subject is emphasized in two ways. One emphasis is based on understanding the influence of the dissipation mechanism on the qualitative behavior of solutions, such as the nonlinear diffusive phenomena caused by damping, and other phenomena (including phase transition) for the case with viscosity and heat diffusion. The second emphasis is to take the systems with the dissipation mechanism as an approach to approximating the corresponding system of quasilinear hyperbolic conservation laws - the zero-limit relaxation, or the zero-limit viscosity, and the related topic of nonlinear stability of waves.
Systems Of Conservation Laws 1
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Author : Denis Serre
language : en
Publisher: Cambridge University Press
Release Date : 1999-05-27
Systems Of Conservation Laws 1 written by Denis Serre and has been published by Cambridge University Press this book supported file pdf, txt, epub, kindle and other format this book has been release on 1999-05-27 with Mathematics categories.
Systems of conservation laws arise naturally in physics and chemistry. To understand them and their consequences (shock waves, finite velocity wave propagation) properly in mathematical terms requires, however, knowledge of a broad range of topics. This book sets up the foundations of the modern theory of conservation laws, describing the physical models and mathematical methods, leading to the Glimm scheme. Building on this the author then takes the reader to the current state of knowledge in the subject. The maximum principle is considered from the viewpoint of numerical schemes and also in terms of viscous approximation. Small waves are studied using geometrical optics methods. Finally, the initial-boundary problem is considered in depth. Throughout, the presentation is reasonably self-contained, with large numbers of exercises and full discussion of all the ideas. This will make it ideal as a text for graduate courses in the area of partial differential equations.
Hyperbolic Problems Theory Numerics And Applications In 2 Volumes
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Author : Tatsien Li
language : en
Publisher: World Scientific
Release Date : 2012-09-28
Hyperbolic Problems Theory Numerics And Applications In 2 Volumes written by Tatsien Li and has been published by World Scientific this book supported file pdf, txt, epub, kindle and other format this book has been release on 2012-09-28 with Mathematics categories.
This two-volume book is devoted to mathematical theory, numerics and applications of hyperbolic problems. Hyperbolic problems have not only a long history but also extremely rich physical background. The development is highly stimulated by their applications to Physics, Biology, and Engineering Sciences; in particular, by the design of effective numerical algorithms. Due to recent rapid development of computers, more and more scientists use hyperbolic partial differential equations and related evolutionary equations as basic tools when proposing new mathematical models of various phenomena and related numerical algorithms.This book contains 80 original research and review papers which are written by leading researchers and promising young scientists, which cover a diverse range of multi-disciplinary topics addressing theoretical, modeling and computational issues arising under the umbrella of ';Hyperbolic Partial Differential Equations';. It is aimed at mathematicians, researchers in applied sciences and graduate students.
Numerical Methods For Conservation Laws
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Author : LEVEQUE
language : en
Publisher: Birkhäuser
Release Date : 2013-11-11
Numerical Methods For Conservation Laws written by LEVEQUE and has been published by Birkhäuser this book supported file pdf, txt, epub, kindle and other format this book has been release on 2013-11-11 with Science categories.
These notes developed from a course on the numerical solution of conservation laws first taught at the University of Washington in the fall of 1988 and then at ETH during the following spring. The overall emphasis is on studying the mathematical tools that are essential in de veloping, analyzing, and successfully using numerical methods for nonlinear systems of conservation laws, particularly for problems involving shock waves. A reasonable un derstanding of the mathematical structure of these equations and their solutions is first required, and Part I of these notes deals with this theory. Part II deals more directly with numerical methods, again with the emphasis on general tools that are of broad use. I have stressed the underlying ideas used in various classes of methods rather than present ing the most sophisticated methods in great detail. My aim was to provide a sufficient background that students could then approach the current research literature with the necessary tools and understanding. vVithout the wonders of TeX and LaTeX, these notes would never have been put together. The professional-looking results perhaps obscure the fact that these are indeed lecture notes. Some sections have been reworked several times by now, but others are still preliminary. I can only hope that the errors are not too blatant. Moreover, the breadth and depth of coverage was limited by the length of these courses, and some parts are rather sketchy.