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.

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.