Lecture Notes on the Discretization of the Boltzmann Equation

Description

3.1 Introduction3.2 Discrete Models with Multiple Collisions; 3.3 Macroscopic Description; 3.4 Boundary Conditions for Discrete Models; 3.5 Conclusion; 3.6 References; Chapter 4. Discretization of the Boltzmann Equation and the Semicontinuous Model; 4.1 Introduction; 4.2 Splitting and Energy Formulation; 4.3 Working in a Finite Energy Interval; 4.4 Energy Discretization and Kinetic Model; 4.5 Conservation and Euler Equations for the Discretized Model; 4.6 Energy Formulation of the Collision Dynamics; 4.7 Concluding Remarks; 4.8 References; Chapter 5. Semi-continuous Extended Kinetic Theory. 5.1 Introduction5.2 Continuous Kinetic Equations; 5.3 Semi-continuous Kinetic Equations; 5.4 Treatment of the Remaining Continuous Variables; 5.5 Relaxational Behaviour; 5.6 Applications; 5.7 Conclusion; 5.8 References; Chapter 6. Steady Kinetic Boundary Value Problems; 6.1 Introduction; 6.2 Discrete kinetic models; 6.3 FEM for the semi discrete steady Boltzmann equation; 6.4 Numerical results; 6.5 References; Chapter 7. Computational Methods and Fast Algorithms for Boltzmann Equations; 7.1 Introduction; 7.2 A one-dimensional example; 7.3 The multi-dimensional case; 7.4 References. Chapter 8. Discrete Velocity Models and Dynamical Systems8.1 Introduction; 8.2 Statement of the Problem; 8.3 DVM as Dynamical System; 8.4 Linearized Equation; 8.5 Indices of Stationary Points; 8.6 Applications to Half-Space Problems; 8.7 Concluding Remarks; 8.8 References; Chapter 9. Numerical Method for the Compton Scattering Operator; 9.1 Introduction; 9.2 Quantum Boltzmann and Kompaneets Equation; 9.3 Semidiscretization; 9.4 Time Discretization for the Kompaneets Equation; 9.5 Numerical Tests for the Kompaneets Schemes; 9.6 Conclusions; 9.7 References. Chapter 10. Discrete Models of the Boltzmann Equation in Quantum Optics and Arbitrary Partition of the Velocity Space10.1 Introduction; 10.2 Laser-induced thermal acoustics: A discrete kinetic approach; 10.3 Modified discrete Boltzmann equation; 10.4 Scaling of discrete velocity models; 10.5 References; List of Contributors. 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.  Read more... Abstract: 3.1 Introduction3.2 Discrete Models with Multiple Collisions; 3.3 Macroscopic Description; 3.4 Boundary Conditions for Discrete Models; 3.5 Conclusion; 3.6 References; Chapter 4. Discretization of the Boltzmann Equation and the Semicontinuous Model; 4.1 Introduction; 4.2 Splitting and Energy Formulation; 4.3 Working in a Finite Energy Interval; 4.4 Energy Discretization and Kinetic Model; 4.5 Conservation and Euler Equations for the Discretized Model; 4.6 Energy Formulation of the Collision Dynamics; 4.7 Concluding Remarks; 4.8 References; Chapter 5. Semi-continuous Extended Kinetic Theory. 5.1 Introduction5.2 Continuous Kinetic Equations; 5.3 Semi-continuous Kinetic Equations; 5.4 Treatment of the Remaining Continuous Variables; 5.5 Relaxational Behaviour; 5.6 Applications; 5.7 Conclusion; 5.8 References; Chapter 6. Steady Kinetic Boundary Value Problems; 6.1 Introduction; 6.2 Discrete kinetic models; 6.3 FEM for the semi discrete steady Boltzmann equation; 6.4 Numerical results; 6.5 References; Chapter 7. Computational Methods and Fast Algorithms for Boltzmann Equations; 7.1 Introduction; 7.2 A one-dimensional example; 7.3 The multi-dimensional case; 7.4 References. Chapter 8. Discrete Velocity Models and Dynamical Systems8.1 Introduction; 8.2 Statement of the Problem; 8.3 DVM as Dynamical System; 8.4 Linearized Equation; 8.5 Indices of Stationary Points; 8.6 Applications to Half-Space Problems; 8.7 Concluding Remarks; 8.8 References; Chapter 9. Numerical Method for the Compton Scattering Operator; 9.1 Introduction; 9.2 Quantum Boltzmann and Kompaneets Equation; 9.3 Semidiscretization; 9.4 Time Discretization for the Kompaneets Equation; 9.5 Numerical Tests for the Kompaneets Schemes; 9.6 Conclusions; 9.7 References. Chapter 10. Discrete Models of the Boltzmann Equation in Quantum Optics and Arbitrary Partition of the Velocity Space10.1 Introduction; 10.2 Laser-induced thermal acoustics: A discrete kinetic approach; 10.3 Modified discrete Boltzmann equation; 10.4 Scaling of discrete velocity models; 10.5 References; List of Contributors. 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 Content: Preface Chapter 1. From the Boltzmann Equation to Discretized Kinetic Models 1.1 Introduction 1.2 The Nonlinear Boltzmann Equation 1.3 The Discrete and Semicontinuous Boltzmann Equation 1.4 Plan of the Lecture Notes 1.5 References Chapter 2. Discrete Velocity Models for Gas Mixtures 2.1 Introduction 2.2 DVM for mixtures 2.3 Models with a finite number of velocities and the problem of spurious invariants 2.4 Constructing DVM with arbitrarily many velocities 2.5 Concluding remarks 2.6 References Chapter 3. Discrete Velocity Models with Multiple Collisions.