Overview

This study consists of investigations into the possibility of the numerical schemes of the direct method for solving the Boltzmann equation. Both deterministic and Monte Carlo procedures are considered to evaluate the collision integrals. The main mathematical tool is the conservative splitting method on the basis of which, a set of classical and new problems are solved to study nonequilibrium gas flows. This monograph differs from other books in the same field, because, for example the book by G.A. Bird is concerned with the approach of simulation of rarefied gas flows and the book by C. Cercignani deals with the classical kinetic theory issues and describes mainly the analytical and engineering methods for solving the Boltzmann equation. Our book is devoted to the numerical direct solving of the Boltzmann equation. The intended level of readership are graduate and postgraduate students and researchers. This book can be used by readers as the mathematical apparatus to numerical study of complex problems of nonequilibrium gas flows.

Full Product Details

Author:   V.V. Aristov
Publisher:   Springer
Imprint:   Springer
Edition:   2001 ed.
Volume:   60
Dimensions:   Width: 15.50cm , Height: 1.90cm , Length: 23.50cm
Weight:   1.390kg
ISBN:  

9780792368311

ISBN 10:   0792368312
Pages:   302
Publication Date:   31 January 2001
Audience:   College/higher education ,  Professional and scholarly ,  Undergraduate ,  Postgraduate, Research & Scholarly
Format:   Hardback
Publisher's Status:   Active
Availability:   In Print  
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Table of Contents

1 The Boltzmann Equation as a Physical and Mathematical Model.- 1.1 Different mathematical forms of the kinetic equation.- 1.2 Peculiarities of kinetic approach for describing physical properties.- 1.3 Formulation of problems and boundary conditions.- 1.4 The forms of the Boltzmann equations in some physical cases.- References.- 2 Survey of Mathematical Approaches to Solving the Boltzmann Equation.- 2.1 General notes on classification of methods.- 2.2 Methods combining analytical and numerical features. Some partial solutions.- 2.3 Approaches based on kinetic models.- 2.4 Numerical simulation methods.- 2.5 Direct simulation Monte Carlo methods.- 2.6 Methods of direct integration.- 2.7 Comparison of direct integration and direct simulation.- References.- 3 Main Features of the Direct Numerical Approaches.- 3.1 Discrete velocities and approximation in velocity space.- 3.2 Approximation in physical space. Finite-difference schemes and iterations.- 3.3 Splitting method.- 3.4 Finite volume scheme.- 3.5 Evaluation of the collision integrals by Monte Carlo technique.- 3.6 Quasi Monte Carlo technique.- References.- 4 Deterministic (Regular) Method for Solving the Boltzmann Equation.- 4.1 General features of the method.- 4.2 Approach to approximation of the collision integrals. Integration over velocity space.- 4.3 Exact evaluation of integrals over impact parameters.- 4.4 Approximation of the collision integrals by quadratic form with constant coefficients.- 4.5 Simplification for velocity space in the case of isotropic symmetry.- References.- 5 Construction of Conservative Scheme for the Kinetic Equation.- 5.1 Different definitions of conservativity.- 5.2 Conservative splitting method.- 5.3 Characteristics and advantages of the conservative schemes.- 5.4 Practical verificationof the method.- 5.5 Conservative method for gas mixtures.- References.- 6 Parallel Algorithms for the Kinetic Equation.- 6.1 Parallel implementation for the direct methods.- 6.2 Several parallel algorithms.- 6.3 Examples of parallel applications of the algorithms.- References.- 7 Application of the Conservative Splitting Method for Investigating Near Continuum Gas Flows.- 7.1 Some approaches to solving the Boltzmann equation for weakly rarefied gas.- 7.2 Asymptotic kinetic schemes approximating the Euler and Navier-Stokes equations.- 7.3 Schemes for flows at low Knudsen numbers.- References.- 8 Study of Uniform Relaxation in Kinetic Gas Theory.- 8.1 Spatially uniform (homogeneous) relaxation problem.- 8.2 Obtaining the test solutions for isotropic relaxation.- 8.3 Some examples of the relaxation problem solutions.- 8.4 Uniform relaxation for gas mixtures.- References.- 9 Nonuniform Relaxation Problem as a Basic Model for Description of Open Systems.- 9.1 Formulation of the problem and solution methods.- 9.2 Nonclassical behavior of macroscopic parameters.- 9.3 Behavior of the distribution function and macroscopic parameters.- 9.4 Possible entropy decrease.- 9.5 Some generalizations.- References.- 10 One-Dimensional Kinetic Problems.- 10.1 The problem of heat transfer.- 10.2 Shock wave structure.- 10.3 Flow in the field of an external force.- 10.4 Recondensation of a mixture in a force field.- References.- 11 Multi-Dimensional Problems. Study of Free Jet Flows.- 11.1 Possibilities of direct integration approaches for studying multi-dimensional problems.- 11.2 Formulation of the problem and numerical scheme.- 11.3 Free plane jet.- 11.4 Axisymmetric and three-dimensional free jet flows.- References.- 12 The Boltzmann Equation and the Description of Unstable Flows.- 12.1 Main notions.- 12.2 Boltzmann and Navier-Stokes description.- 12.3 Mathematical apparatus.- 12.4 Some results of numerical modelling..- References.- 13 Solutions of some Multi-Dimensional Problems.- 13.1 Unsteady problem of a shock wave reflection from a wedge.- 13.2 Solution for focusing of a shock wave.- 13.3 Study of flows in elements of cryovacuum devices.- 13.4 Flows in the vacuum cryomodulus.- 13.5 Two-component mixture flows with cryocondensation.- References.- 14 Special Hypersonic Flows and Flows with Very High Temperatures.- 14.1 Special hypersonic flows.- 14.2 Unsteady flows caused by a powerful point discharge of a finite gaseous mass.- 14.3 Asymptotic solution at t ? 0.- 14.4 Numerical analysis. Asymptotic solution at t ? ?.- 14.5 Scattering of impulsive molecular beam.- References.

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