Direct Methods for Solving the Boltzmann Equation and Study of Nonequilibrium Flows

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. Language: English