This work 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 written by an applied mathematician who has been active in the field, and whose scientific contributions are well recognized by the scientific community.
Author(s): N. Bellomo, Renee Gatignol
Series: Series on Advances in Mathematics for Applied Sciences
Year: 2003
Language: English
Pages: 316
9812382259......Page 1
CONTENTS......Page 8
Preface......Page 10
1.1 Introduction......Page 12
1.2 The Nonlinear Boltzmann Equation......Page 13
1.3 The Discrete and Semicontinuous Boltzmann Equation......Page 19
1.4 Plan of the Lecture Notes......Page 24
1.5 References......Page 25
2.1 Introduction......Page 28
2.2 DVM for mixtures......Page 29
2.3 Models with a finite number of velocities and the problem of spurious invariants......Page 32
2.4 Constructing DVM with arbitrarily many velocities......Page 34
2.5 Concluding remarks......Page 36
2.6 References......Page 37
3.1 Introduction......Page 40
3.2 Discrete Models with Multiple Collisions......Page 42
3.3 Macroscopic Description......Page 48
3.4 Boundary Conditions for Discrete Models......Page 59
3.5 Conclusion......Page 65
3.6 References......Page 66
4.1 Introduction......Page 70
4.2 Splitting and Energy Formulation......Page 72
4.3 Working in a Finite Energy Interval......Page 78
4.4 Energy Discretization and Kinetic Model......Page 87
4.5 Conservation and Euler Equations for the Discretized Model......Page 90
4.6 Energy Formulation of the Collision Dynamics......Page 101
4.7 Concluding Remarks......Page 103
4.8 References......Page 105
5.1 Introduction......Page 108
5.2 Continuous Kinetic Equations......Page 110
5.3 Semi-continuous Kinetic Equations......Page 113
5.4 Treatment of the Remaining Continuous Variables......Page 121
5.5 Relaxational Behaviour......Page 125
5.6 Applications......Page 130
5.7 Conclusion......Page 141
5.8 References......Page 142
6.1 Introduction......Page 144
6.2 Discrete kinetic models......Page 145
6.3 FEM for the semi discrete steady Boltzmann equation......Page 158
6.4 Numerical results......Page 163
6.5 References......Page 167
7.1 Introduction......Page 170
7.2 A one-dimensional example......Page 174
7.3 The multi-dimensional case......Page 185
7.4 References......Page 210
8.1 Introduction......Page 214
8.2 Statement of the Problem......Page 215
8.3 DVM as Dynamical System......Page 218
8.4 Linearized Equation......Page 221
8.5 Indices of Stationary Points......Page 226
8.6 Applications to Half-Space Problems......Page 229
8.7 Concluding Remarks......Page 231
8.8 References......Page 232
9.1 Introduction......Page 234
9.2 Quantum Boltzmann and Kompaneets Equation......Page 235
9.3 Semidiscretization......Page 246
9.4 Time Discretization for the Kompaneets Equation......Page 256
9.5 Numerical Tests for the Kompaneets Schemes......Page 259
9.6 Conclusions......Page 263
9.7 References......Page 268
10.1 Introduction......Page 270
10.2 Laser-induced thermal acoustics: A discrete kinetic approach......Page 272
10.3 Modified discrete Boltzmann equation......Page 285
10.4 Scaling of discrete velocity models......Page 300
10.5 References......Page 307
List of Contributors......Page 310
