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Author : Alexander V. Bobylev Publisher : Walter de Gruyter GmbH & Co KG Page : 289 pages File Size : 39,47 MB Release : 2024-09-23 Category : Mathematics ISBN : 3110550148
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.
This book presents contributions on the following topics: discretization methods in the velocity and space, analysis of the conservation properties, asymptotic convergence to the continuous equation when the number of velocities tends to infinity, and application of discrete models. It consists of ten chapters. Each chapter is written by applied mathematicians who have been active in the field, and whose scientific contributions are well recognized by the scientific community. Contents: From the Boltzmann Equation to Discretized Kinetic Models (N Bellomo & R Gatignol); Discrete Velocity Models for Gas Mixtures (C Cercignani); Discrete Velocity Models with Multiple Collisions (R Gatignol); Discretization of the Boltzmann Equation and the Semicontinuous Model (L Preziosi & L Rondoni); Semi-continuous Extended Kinetic Theory (W Koller); Steady Kinetic Boundary Value Problems (H Babovsky et al.); Computational Methods and Fast Algorithms for the Boltzmann Equation (L Pareschi); Discrete Velocity Models and Dynamical Systems (A Bobylev & N Bernhoff); Numerical Method for the Compton Scattering Operator (C Buet & S Cordier); Discrete Models of the Boltzmann Equation in Quantum Optics and Arbitrary Partition of the Velocity Space (F Schrrer). Readership: Higher level postgraduates in applied mathematics.
Handbook on Numerical Methods for Hyperbolic Problems: Applied and Modern Issues details the large amount of literature in the design, analysis, and application of various numerical algorithms for solving hyperbolic equations that has been produced in the last several decades. This volume provides concise summaries from experts in different types of algorithms, so that readers can find a variety of algorithms under different situations and become familiar with their relative advantages and limitations. Provides detailed, cutting-edge background explanations of existing algorithms and their analysis Presents a method of different algorithms for specific applications and the relative advantages and limitations of different algorithms for engineers or those involved in applications Written by leading subject experts in each field, the volumes provide breadth and depth of content coverage
Boltzmann and Vlasov equations played a great role in the past and still play an important role in modern natural sciences, technique and even philosophy of science. Classical Boltzmann equation derived in 1872 became a cornerstone for the molecular-kinetic theory, the second law of thermodynamics (increasing entropy) and derivation of the basic hydrodynamic equations. After modifications, the fields and numbers of its applications have increased to include diluted gas, radiation, neutral particles transportation, atmosphere optics and nuclear reactor modelling. Vlasov equation was obtained in 1938 and serves as a basis of plasma physics and describes large-scale processes and galaxies in astronomy, star wind theory. This book provides a comprehensive review of both equations and presents both classical and modern applications. In addition, it discusses several open problems of great importance. Reviews the whole field from the beginning to today Includes practical applications Provides classical and modern (semi-analytical) solutions
This book is concerned with the methods of solving the nonlinear Boltz mann equation and of investigating its possibilities for describing some aerodynamic and physical problems. This monograph is a sequel to the book 'Numerical direct solutions of the kinetic Boltzmann equation' (in Russian) which was written with F. G. Tcheremissine and published by the Computing Center of the Russian Academy of Sciences some years ago. The main purposes of these two books are almost similar, namely, the study of nonequilibrium gas flows on the basis of direct integration of the kinetic equations. Nevertheless, there are some new aspects in the way this topic is treated in the present monograph. In particular, attention is paid to the advantages of the Boltzmann equation as a tool for considering nonequi librium, nonlinear processes. New fields of application of the Boltzmann equation are also described. Solutions of some problems are obtained with higher accuracy. Numerical procedures, such as parallel computing, are in vestigated for the first time. The structure and the contents of the present book have some com mon features with the monograph mentioned above, although there are new issues concerning the mathematical apparatus developed so that the Boltzmann equation can be applied for new physical problems. Because of this some chapters have been rewritten and checked again and some new chapters have been added.
An invaluable instrument for gaining a wide-ranging perspective on the latest developments in mathematical aspects of scientific computing, discovering new applications and the most recent developments in long-standing applications. Provides an insight into the state of the art of Numerical Mathematics and, more generally, into the field of Advanced Applications.
Author : Alexander V. Bobylev Publisher : Walter de Gruyter GmbH & Co KG Page : 275 pages File Size : 49,53 MB Release : 2020-10-12 Category : Mathematics ISBN : 3110550172
The series is devoted to the publication of high-level monographs and specialized graduate texts which cover the whole spectrum of applied mathematics, including its numerical aspects. The focus of the series is on the interplay between mathematical and numerical analysis, and also on its applications to mathematical models in the physical and life sciences. The aim of the series is to be an active forum for the dissemination of up-to-date information in the form of authoritative works that will serve the applied mathematics community as the basis for further research. Editorial Board Rémi Abgrall, Universität Zürich, Switzerland José Antonio Carrillo de la Plata, University of Oxford, UK Jean-Michel Coron, Université Pierre et Marie Curie, Paris, France Athanassios S. Fokas, Cambridge University, UK Irene Fonseca, Carnegie Mellon University, Pittsburgh, USA
The focus of this thesis is to investigate a conservative spectral method for solving Fokker-Planck-Landau type (F. P. L.) equations as a model for plasmas, when coupled to the Vlasov-Poisson equation in the mean-field limit, modelling particle interactions extending from Coulomb to hard sphere potentials. This study will range from numerical examples, that emphasise the strength and accuracy of the method, to a rigorous proof showing that approximations from the numerical scheme converge to analytical solutions. In particular, two sets of novel simulations are included. The first presents benchmark results of decay rates to statistical equilibrium in transport plasma models for Coulomb particle interactions, as well as with Maxwell type and hard sphere interactions. The other studies the essentially unexplored phenomenon of the plasma sheath for Coulomb interactions, exhibiting the formation of a strong field due to charge separation. These topics will be arranged in three major projects: 1. Entropy decay rates for the conservative spectral scheme modelling Fokker-Planck-Landau type flows in the mean field limit. Benchmark simulations of decay rates to statistical equilibrium are created for F.P.L. equations associated to Coulomb particle interactions, as well as with Maxwell type and hard sphere interactions. The qualitative decay to the equilibrium Maxwell-Boltzmann distribution is studied in detail through relative entropy for all three types of particle interactions by means of a conservative hybrid spectral and discontinuous Galerkin scheme, adapted from Chenglong Zhang’s thesis in 2014. More precisely, the Coulomb case shows that there is a degenerate spectrum, with a decay rate close to the law of two thirds predicted by upper estimates in work by Strain and Guo in 2006, while the Maxwell type and hard sphere examples both exhibit a spectral gap as predicted by Desvillettes and Villani in 2000. Such decay rate behaviour indicates that the analytical estimates for the Coulomb case is sharp while, still to this date, there is no analytical proof of sharp degenerate spectral behaviour for the F.P.L. operator. Simulations are presented, both for the space-homogeneous case of just particle potential interactions and the space-inhomogeneous case with the mean field coupling through the Poisson equation for total charges in periodic domains. New explicit derivations of spectral collisional weights are presented in the case of Maxwell type and hard sphere interactions and the stability of all three scenarios, including Coulomb interactions, is investigated. 2. Convergence and error estimates for the conservative spectral method for Fokker-Planck-Landau equations. Error estimates are rigorously derived for a semi-discrete version of the conservative spectral method for approximating the space-homogeneous F.P.L. equation associated to hard potentials. The analysis included shows that the semi-discrete problem has a unique solution with bounded moments. In addition, the derivatives of such a solution up to any order also remain bounded in L2 spaces globally time, under certain conditions. These estimates, combined with spectral projection control, are enough to obtain error estimates to the analytical solution and convergence to equilibrium states. It should be noted that this is the first time that an error estimate has been produced for any numerical method which approximates F.P.L. equations associated to any range of potentials. 3. Modelling charge separation with the Landau equation. A model for the plasma sheath is investigated using the space-inhomogeneous linear Landau equation (namely, the F.P.L. equation associated Coulomb interactions), modelling interactions between positive and negative particles. Some theory has been established for the plasma sheath, but this is the first time that an attempt has been made to simulate it with the Landau equation. The particular design of the kinetic model is described and an attempt made to capture physical phenomena associated to separation of charges. Several parameters within the model are varied to try and explain the creation of the sheaths
Progress in Computational Physics is an e-book series devoted to recent research trends in computational physics. It contains chapters contributed by outstanding experts of modeling of physical problems. The series focuses on interdisciplinary computational perspectives of current physical challenges, new numerical techniques for the solution of mathematical wave equations and describes certain real-world applications. With the help of powerful computers and sophisticated methods of numerical mathematics it is possible to simulate many ultramodern devices, e.g. photonic crystals structures, semiconductor nanostructures or fuel cell stacks devices, thus preventing expensive and longstanding design and optimization in the laboratories. In this book series, research manuscripts are shortened as single chapters and focus on one hot topic per volume. Engineers, physicists, meteorologists, etc. and applied mathematicians can benefit from the series content. Readers will get a deep and active insight into state-of-the art modeling and simulation techniques of ultra-modern devices and problems. The third volume - Novel Trends in Lattice Boltzmann Methods - Reactive Flow, Physicochemical Transport and Fluid-Structure Interaction - contains 10 chapters devoted to mathematical analysis of different issues related to the lattice Boltzmann methods, advanced numerical techniques for physico-chemical flows, fluid structure interaction and practical applications of these phenomena to real world problems.