Methods of direct solving the Boltzmann equation and study of nonequilibrium flows
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The outstanding points of our book consist 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 the first (as we know) monograph which is devoted to the numerical direct solving of the Boltzmann equation. The intended level of readership are graduate and postgraduate students and researches. This book can be used by the target groups as the mathematical apparatus to numerical study of complex problems of nonequilibrium gas flows. Booknews Concerned with the methods of solving the nonlinear Baltzmann equation and investigating its possibilities for describing some aerodynamical and physical problems, the main purpose of this work is the study of nonequilibrium gas flows on the basis of the direct integration of the kinetic equations. The monograph is a revision of Aristov's and Thcherenessine's (in Russian), with the main difference being more attention to the advantages of the Boltzmann equation as a tool for considering nonlinear, nonequilibrium processes. Annotation c. Book News, Inc., Portland, OR (booknews.com)
PrefaceixIntroductionxiiiReferencesxvii1The Boltzmann Equation as a Physical and Mathematical Model11.1Different mathematical forms of the kinetic equation11.2Peculiarities of kinetic approach for describing physical properties61.3Formulation of problems and boundary conditions101.4The forms of the Boltzmann equations in some physical cases13References212Survey of Mathematical Approaches to Solving the Boltzmann Equation232.1General notes on classification of methods232.2Methods combining analytical and numerical features. Some partial solutions252.3Approaches based on kinetic models272.4Numerical simulation methods292.5Direct simulation Monte Carlo methods302.6Methods of direct integration312.7Comparison of direct integration and direct simulation33References393Main Features of the Direct Numerical Approaches453.1Discrete velocities and approximation in velocity space453.2Approximation in physical space. Finite-difference schemes and iterations493.3Splitting method513.4Finite volume scheme563.5Evaluation of the collision integrals by Monte Carlo technique583.6Quasi Monte Carlo technique61References674Deterministic (Regular) Method for Solving the Boltzmann Equation694.1General features of the method694.2Approach to approximation of the collision integrals. Integration over velocity space694.3Exact evaluation of integrals over impact parameters704.4Approximation of the collision integrals by quadratic form with constant coefficients754.5Simplification for velocity space in the case of isotropic symmetry77References835Construction of Conservative Scheme for the Kinetic Equation855.1Different definitions of conservativity855.2Conservative splitting method875.3Characteristics and advantages of the conservative schemes935.4Practical verification of the method985.5Conservative method for gas mixtures103References1076Parallel Algorithms for the Kinetic Equation1096.1Parallel implementation for the direct methods1096.2Several parallel algorithms1116.3Examples of parallel applications of the algorithms113References1197Application of the Conservative Splitting Method for Investigating Near Continuum Gas Flows1217.1Some approaches to solving the Boltzmann equation for weakly rarefied gas1217.2Asymptotic kinetic schemes approximating the Euler and Navier-Stokes equations1247.3Schemes for flows at low Knudsen numbers131References1378Study of Uniform Relaxation in Kinetic Gas Theory1398.1Spatially uniform (homogeneous) relaxation problem1398.2Obtaining the test solutions for isotropic relaxation1408.3Some examples of the relaxation problem solutions1468.4Uniform relaxation for gas mixtures148References1539Nonuniform Relaxation Problem as a Basic Model for Description of Open Systems1559.1Formulation of the problem and solution methods1559.2Nonclassical behavior of macroscopic parameters1599.3Behavior of the distribution function and macroscopic parameters1649.4Possible entropy decrease1679.5Some generalizations171References17910One-Dimensional Kinetic Problems18110.1The problem of heat transfer18110.2Shock wave structure18810.3Flow in the field of an external force19710.4Recondensation of a mixture in a force field203References20711Multi-Dimensional Problems. Study of Free Jet Flows21111.1Possibilities of direct integration approaches for studying multi-dimensional problems21111.2Formulation of the problem and numerical scheme21211.3Free plane jet21411.4Axisymmetric and three-dimensional free jet flows215References22512The Boltzmann Equation and the Description of Unstable Flows22712.1Main notions22712.2Boltzmann and Navier-Stokes description22812.3Mathematical apparatus23012.4Some results of numerical modelling231References23913Solutions of Some Multi-Dimensional Problems24113.1Unsteady problem of a shock wave reflection from a wedge24113.2Solution for focusing of a shock wave25013.3Study of flows in elements of cryovacuum devices25413.4Flows in the vacuum cryomodulus26013.5Two-component mixture flows with cryocondensation263References26914Special Hypersonic Flows and Flows with Very High Temperatures27114.1Special hypersonic flows27114.2Unsteady flows caused by a powerful point discharge of a finite gaseous mass27414.3Asymptotic solution at t [right arrow] 027714.4Numerical analysis. Asymptotic solution at t [right arrow] [infinity]28114.5Scattering of impulsive molecular beam286References293
