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Numerical Analysis Of Kinetic Equations And Their Hydrodynamical Limit
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Book Synopsis Coupling of Kinetic Equations and Their Hydrodynamic Limits by :
Download or read book Coupling of Kinetic Equations and Their Hydrodynamic Limits written by and published by . This book was released on 2001 with total page 25 pages. Available in PDF, EPUB and Kindle. Book excerpt: During the last two years, the author continued his efforts to reach the above objectives. In particular he focused on the following: Further numerical and theoretical studies of the coupling of Boltzmann equation and its hydrodynamic limits. Numerical and theoretical analysis of the coupling of kinetic equations and their hydrodynamic limits for particular kinetic models. (This gives more insight into the coupling method for the full Boltzmann model). Numerical and mathematical studies of the coupling of BCE and ES-BCE with their hydrodynamic limits. (The study of BCE and ES-BCE led the author to replace in the coupling of Boltzmann equations and their hydrodynamic limits, the Boltzmann equations by ES-BCE model. The author plan now is to show that this resulting coupling is optimal and can be used for the numerical simulation of flows around full airplanes at high altitudes). This study is subdivided into two parts In the monoatomic case. The author is developing and validating his coupling using Direct Simulation Monte Carlo method for the kinetic model. He is also planning on developing a deterministic approach for the kinetic model. This will cut the computer cost considerably and allows for the computations around full airplanes. In the polyatomic case, the author is developing and validating his coupling using Direct Simulation Monte Carlo method for the kinetic model when an internal energy variable is used.
Book Synopsis Modeling and Computational Methods for Kinetic Equations by : Pierre Degond
Download or read book Modeling and Computational Methods for Kinetic Equations written by Pierre Degond and published by Springer Science & Business Media. This book was released on 2012-12-06 with total page 360 pages. Available in PDF, EPUB and Kindle. Book excerpt: In recent years kinetic theory has developed in many areas of the physical sciences and engineering, and has extended the borders of its traditional fields of application. This monograph is a self-contained presentation of such recently developed aspects of kinetic theory, as well as a comprehensive account of the fundamentals of the theory. Emphasizing modeling techniques and numerical methods, the book provides a unified treatment of kinetic equations not found in more focused works. Specific applications presented include plasma kinetic models, traffic flow models, granular media models, and coagulation-fragmentation problems. The work may be used for self-study, as a reference text, or in graduate-level courses in kinetic theory and its applications.
Book Synopsis Kinetic Theory and Fluid Dynamics by : Yoshio Sone
Download or read book Kinetic Theory and Fluid Dynamics written by Yoshio Sone and published by Springer Science & Business Media. This book was released on 2012-12-06 with total page 358 pages. Available in PDF, EPUB and Kindle. Book excerpt: This monograph is intended to provide a comprehensive description of the rela tion between kinetic theory and fluid dynamics for a time-independent behavior of a gas in a general domain. A gas in a steady (or time-independent) state in a general domain is considered, and its asymptotic behavior for small Knudsen numbers is studied on the basis of kinetic theory. Fluid-dynamic-type equations and their associated boundary conditions, together with their Knudsen-layer corrections, describing the asymptotic behavior of the gas for small Knudsen numbers are presented. In addition, various interesting physical phenomena derived from the asymptotic theory are explained. The background of the asymptotic studies is explained in Chapter 1, accord ing to which the fluid-dynamic-type equations that describe the behavior of a gas in the continuum limit are to be studied carefully. Their detailed studies depending on physical situations are treated in the following chapters. What is striking is that the classical gas dynamic system is incomplete to describe the behavior of a gas in the continuum limit (or in the limit that the mean free path of the gas molecules vanishes). Thanks to the asymptotic theory, problems for a slightly rarefied gas can be treated with the same ease as the corresponding classical fluid-dynamic problems. In a rarefied gas, a temperature field is di rectly related to a gas flow, and there are various interesting phenomena which cannot be found in a gas in the continuum limit.
Book Synopsis From Kinetic Models to Hydrodynamics by : Matteo Colangeli
Download or read book From Kinetic Models to Hydrodynamics written by Matteo Colangeli and published by Springer Science & Business Media. This book was released on 2013-03-25 with total page 102 pages. Available in PDF, EPUB and Kindle. Book excerpt: From Kinetic Models to Hydrodynamics serves as an introduction to the asymptotic methods necessary to obtain hydrodynamic equations from a fundamental description using kinetic theory models and the Boltzmann equation. The work is a survey of an active research area, which aims to bridge time and length scales from the particle-like description inherent in Boltzmann equation theory to a fully established “continuum” approach typical of macroscopic laws of physics.The author sheds light on a new method—using invariant manifolds—which addresses a functional equation for the nonequilibrium single-particle distribution function. This method allows one to find exact and thermodynamically consistent expressions for: hydrodynamic modes; transport coefficient expressions for hydrodynamic modes; and transport coefficients of a fluid beyond the traditional hydrodynamic limit. The invariant manifold method paves the way to establish a needed bridge between Boltzmann equation theory and a particle-based theory of hydrodynamics. Finally, the author explores the ambitious and longstanding task of obtaining hydrodynamic constitutive equations from their kinetic counterparts. The work is intended for specialists in kinetic theory—or more generally statistical mechanics—and will provide a bridge between a physical and mathematical approach to solve real-world problems.
Book Synopsis Kinetic Equations by : Alexander V. Bobylev
Download or read book Kinetic Equations written by Alexander V. Bobylev and published by Walter de Gruyter GmbH & Co KG. This book was released on 2020-10-12 with total page 260 pages. Available in PDF, EPUB and Kindle. Book excerpt: This two-volume monograph is a comprehensive and up-to-date presentation of the theory and applications of kinetic equations. The first volume covers many-particle dynamics, Maxwell models of the Boltzmann equation (including their exact and self-similar solutions), and hydrodynamic limits beyond the Navier-Stokes level.
Book Synopsis The Cauchy Problem in Kinetic Theory by : Robert T. Glassey
Download or read book The Cauchy Problem in Kinetic Theory written by Robert T. Glassey and published by SIAM. This book was released on 1996-01-01 with total page 246 pages. Available in PDF, EPUB and Kindle. Book excerpt: Studies the basic equations of kinetic theory in all of space, and contains up-to-date, state-of-the-art treatments of initial-value problems for the major kinetic equations. This is the only existing book to treat Boltzmann-type problems and Vlasov-type problems together. Although describing very different phenomena, these equations share the same streaming term.
Book Synopsis Hydrodynamic Limits of the Boltzmann Equation by : Laure Saint-Raymond
Download or read book Hydrodynamic Limits of the Boltzmann Equation written by Laure Saint-Raymond and published by Springer Science & Business Media. This book was released on 2009-03-26 with total page 203 pages. Available in PDF, EPUB and Kindle. Book excerpt: "The material published in this volume comes essentially from a course given at the Conference on "Boltzmann equation and fluidodynamic limits", held in Trieste in June 2006." -- preface.
Book Synopsis Handbook of Differential Equations: Evolutionary Equations by : C.M. Dafermos
Download or read book Handbook of Differential Equations: Evolutionary Equations written by C.M. Dafermos and published by Elsevier. This book was released on 2011-09-22 with total page 653 pages. Available in PDF, EPUB and Kindle. Book excerpt: The material collected in this volume reflects the active present of this area of mathematics, ranging from the abstract theory of gradient flows to stochastic representations of non-linear parabolic PDE's. Articles will highlight the present as well as expected future directions of development of the field with particular emphasis on applications. The article by Ambrosio and Savaré discusses the most recent development in the theory of gradient flow of probability measures. After an introduction reviewing the properties of the Wasserstein space and corresponding subdifferential calculus, applications are given to evolutionary partial differential equations. The contribution of Herrero provides a description of some mathematical approaches developed to account for quantitative as well as qualitative aspects of chemotaxis. Particular attention is paid to the limits of cell's capability to measure external cues on the one hand, and to provide an overall description of aggregation models for the slim mold Dictyostelium discoideum on the other. The chapter written by Masmoudi deals with a rather different topic - examples of singular limits in hydrodynamics. This is nowadays a well-studied issue given the amount of new results based on the development of the existence theory for rather general systems of equations in hydrodynamics. The paper by DeLellis addreses the most recent results for the transport equations with regard to possible applications in the theory of hyperbolic systems of conservation laws. Emphasis is put on the development of the theory in the case when the governing field is only a BV function. The chapter by Rein represents a comprehensive survey of results on the Poisson-Vlasov system in astrophysics. The question of global stability of steady states is addressed in detail. The contribution of Soner is devoted to different representations of non-linear parabolic equations in terms of Markov processes. After a brief introduction on the linear theory, a class of non-linear equations is investigated, with applications to stochastic control and differential games. The chapter written by Zuazua presents some of the recent progresses done on the problem of controllabilty of partial differential equations. The applications include the linear wave and heat equations,parabolic equations with coefficients of low regularity, and some fluid-structure interaction models. - Volume 1 focuses on the abstract theory of evolution - Volume 2 considers more concrete probelms relating to specific applications - Volume 3 reflects the active present of this area of mathematics, ranging from the abstract theory of gradient flows to stochastic representations of non-linear PDEs
Book Synopsis Landau Equation, Boltzmann-type Equations, Discrete Models, and Numerical Methods by : Alexander V. Bobylev
Download or read book Landau Equation, Boltzmann-type Equations, Discrete Models, and Numerical Methods written by Alexander V. Bobylev and published by Walter de Gruyter GmbH & Co KG. This book was released on 2024-09-23 with total page 289 pages. Available in PDF, EPUB and Kindle. Book excerpt: This two-volume monograph is a comprehensive and up-to-date presentation of the theory and applications of kinetic equations. The second volume covers discrete velocity models of the Boltzmann equation, results on the Landau equation, and numerical (deterministic and stochastic) methods for the solution of kinetic equations.
Book Synopsis Hydrodynamic Limits and Related Topics by : Shui Feng
Download or read book Hydrodynamic Limits and Related Topics written by Shui Feng and published by American Mathematical Soc.. This book was released on 2000 with total page 153 pages. Available in PDF, EPUB and Kindle. Book excerpt: This book presents the lecture notes and articles from the workshop on hydrodynamic limits held at The Fields Institute (Toronto). The first part of the book contains the notes from the mini-course given by Professor S. R. S. Varadhan. The second part contains research articles reviewing the diverse progress in the study of hydrodynamic limits and related areas. This book offers a comprehensive introduction to the theory and its techniques, including entropy and relative entropy methods, large deviation estimates, and techniques in nongradient systems. This book, especially the lectures of Part I, could be used as a text for an advanced graduate course in hydrodynamic limits and interacting particle systems.
Book Synopsis Modeling and Computational Methods for Kinetic Equations by : Pierre Degond
Download or read book Modeling and Computational Methods for Kinetic Equations written by Pierre Degond and published by . This book was released on 2004 with total page 356 pages. Available in PDF, EPUB and Kindle. Book excerpt: In recent years kinetic theory has developed in many areas of the physical sciences and engineering, and has extended the borders of its traditional fields of application. This monograph is a self-contained presentation of such recently developed aspects of kinetic theory, as well as a comprehensive account of the fundamentals of the theory. Emphasizing modeling techniques and numerical methods, the book provides a unified treatment of kinetic equations not found in more focused works. Specific applications presented include plasma kinetic models, traffic flow models, granular media models, and coagulation-fragmentation problems. The work may be used for self-study, as a reference text, or in graduate-level courses in kinetic theory and its applications.
Book Synopsis Numerical Methods for the Coupling of Multispecies Kinetic and Hydrodynamic Equations by : Yingzhe Shi
Download or read book Numerical Methods for the Coupling of Multispecies Kinetic and Hydrodynamic Equations written by Yingzhe Shi and published by . This book was released on 2009 with total page 79 pages. Available in PDF, EPUB and Kindle. Book excerpt:
Book Synopsis Mathematical Methods for Hydrodynamic Limits by : Anna DeMasi
Download or read book Mathematical Methods for Hydrodynamic Limits written by Anna DeMasi and published by Springer. This book was released on 2006-11-14 with total page 204 pages. Available in PDF, EPUB and Kindle. Book excerpt: Entropy inequalities, correlation functions, couplings between stochastic processes are powerful techniques which have been extensively used to give arigorous foundation to the theory of complex, many component systems and to its many applications in a variety of fields as physics, biology, population dynamics, economics, ... The purpose of the book is to make theseand other mathematical methods accessible to readers with a limited background in probability and physics by examining in detail a few models where the techniques emerge clearly, while extra difficulties arekept to a minimum. Lanford's method and its extension to the hierarchy of equations for the truncated correlation functions, the v-functions, are presented and applied to prove the validity of macroscopic equations forstochastic particle systems which are perturbations of the independent and of the symmetric simple exclusion processes. Entropy inequalities are discussed in the frame of the Guo-Papanicolaou-Varadhan technique and of theKipnis-Olla-Varadhan super exponential estimates, with reference to zero-range models. Discrete velocity Boltzmann equations, reaction diffusion equations and non linear parabolic equations are considered, as limits of particles models. Phase separation phenomena are discussed in the context of Glauber+Kawasaki evolutions and reaction diffusion equations. Although the emphasis is onthe mathematical aspects, the physical motivations are explained through theanalysis of the single models, without attempting, however to survey the entire subject of hydrodynamical limits.
Book Synopsis Analysis of Hydrodynamic Models by : Peter Constantin
Download or read book Analysis of Hydrodynamic Models written by Peter Constantin and published by SIAM. This book was released on 2017-04-25 with total page 67 pages. Available in PDF, EPUB and Kindle. Book excerpt: Analysis of Hydrodynamic Models presents a concise treatment of a number of partial differential equations of hydrodynamic origin, including the incompressible Euler equations, SQG, Boussinesq, incompressible porous medium, and Oldroyd-B. The author?s approach is based on properties of the particle trajectory maps and on analysis of the back-and-forth passage between the Lagrangian and the Eulerian descriptions. This concise, unified approach brings readers up to date on current open problems. This book is intended for graduate students and junior researchers in mathematics. This book is intended for graduate students and junior researchers in mathematics.
Book Synopsis On Hydrodynamic Equations and Their Relation to Kinetic Theory and Statistical Mechanics by : Jianfei Xue (dd)
Download or read book On Hydrodynamic Equations and Their Relation to Kinetic Theory and Statistical Mechanics written by Jianfei Xue (dd) and published by . This book was released on 2017 with total page 123 pages. Available in PDF, EPUB and Kindle. Book excerpt: We establish conditions on the Hamiltonian evolution of interacting molecules that imply hydrodynamic equations at the limit of infinitely many molecules and show that these conditions are satisfied whenever the solutions of the classical equations for N interacting molecules obey uniform in N bounds. We show that this holds when the initial conditions are bounded and the molecule interaction is weak enough at the initial time. We then obtain hydrodynamic equations that coincide with Maxwell's. We then construct explicit examples of spontaneous energy generation and nonuniqueness for the standard compressible Euler system, with and without pressure, agagin by taking limits of Hamiltonian dynamics as the number of molecules increases to infinity. The examples come from rescaling of well-posed, deterministic systems of molecules that either collide elastically or interact via singular pair potentials. We also obtain Percus macroscopic equation as the limit of a sequence of single systems of N hard rods with the number of hard rods going to infinity. Finally, we establish the strict convexity of the pressure as a thermodynamic limit for continuous systems. As a result we show the existence of a local bijection between macroscopic density, velocity, and energy on one hand and thermodynamic parameters on the other, for continuous systems.
Book Synopsis Recent Advances in Kinetic Equations and Applications by : Francesco Salvarani
Download or read book Recent Advances in Kinetic Equations and Applications written by Francesco Salvarani and published by Springer. This book was released on 2021-12-22 with total page 393 pages. Available in PDF, EPUB and Kindle. Book excerpt: The volume covers most of the topics addressed and discussed during the Workshop INdAM "Recent advances in kinetic equations and applications", which took place in Rome (Italy), from November 11th to November 15th, 2019. The volume contains results on kinetic equations for reactive and nonreactive mixtures and on collisional and noncollisional Vlasov equations for plasmas. Some contributions are devoted to the study of phase transition phenomena, kinetic problems with nontrivial boundary conditions and hierarchies of models. The book, addressed to researchers interested in the mathematical and numerical study of kinetic equations, provides an overview of recent advances in the field and future research directions.
Book Synopsis Numerical Methods for the Simulation of Multi-phase and Complex Flow by : T. M. M. Verheggen
Download or read book Numerical Methods for the Simulation of Multi-phase and Complex Flow written by T. M. M. Verheggen and published by Lecture Notes in Control and I. This book was released on 1992 with total page 168 pages. Available in PDF, EPUB and Kindle. Book excerpt: The nine review articles contained here introduce the techniques required touse lattice gas methods for numerical simulations of complex flows. Furthermore, lattice Boltzmann models are studied together with classical numerical techniques. The editors have written an extensive introductionto this exciting new approach to solving practical problems in modelling andsimulating flows. The book addresses numerical analysts and engineers in fluid mechanics, but also graduate students.
