Stochastic Numerics for the Boltzmann Equation (eBook)

Preface 5
List of notations 8
Contents 10
1 Kinetic theory 13
1.1 The Boltzmann equation 13
1.2 Collision transformations 14
1.3 Collision kernels 20
1.4 Boundary conditions 23
1.5 Physical properties of gas flows 24
1.6 Properties of the collision integral 28
1.7 Moment equations 33
1.8 Criterion of local equilibrium 36
1.9 Scaling transformations 40
1.10 Comments and bibliographic remarks 42
2 Related Markov processes 45
2.1 Boltzmann type piecewise-deterministic Markov processes 45
2.2 Heuristic derivation of the limiting equation 53
2.3 Special cases and bibliographic remarks 59
3 Stochastic weighted particle method 77
3.1 The DSMC framework 79
3.2 Free flow part 83
3.3 Collision part 90
3.4 Controlling the number of particles 109
3.5 Comments and bibliographic remarks 144
4 Numerical experiments 159
4.1 Maxwellian initial state 161
4.2 Relaxation of a mixture of two Maxwellians 169
4.3 BKW solution of the Boltzmann equation 183
4.4 Eternal solution of the Boltzmann equation 190
4.5 A spatially one-dimensional example 193
4.6 A spatially two-dimensional example 210
A Auxiliary results 223
A.1 Properties of the Maxwell distribution 223
A.2 Exact relaxation of moments 225
A.3 Properties of the BKW solution 229
A.4 Convergence of random measures 232
A.5 Existence of solutions 235
B Modeling of distributions 240
B.1 General techniques 240
B.2 Uniform distribution on the unit sphere 242
B.3 Directed distribution on the unit sphere 243
B.4 Maxwell distribution 245
B.5 Directed half-space Maxwell distribution 247
B.6 Initial distribution of the BKW solution 250
B.7 Initial distribution of the eternal solution 252
References 254
Index 265