Dieter A. Wolf-Gladrow
Lattice-Gas
Cellular Automata
and Lattice Boltzmann
Models
An Introduction
Springer
Table of Contents
1.
Introduction
1
1.1
Preface
2
1.2
Overview
4
1.3
The basic idea of lattice-gas cellular automata and lattice
Boltzmann models
7
1.3.1
The Navier-Stokes equation
7
1.3.2
The basic idea
9
1.3.3
Top-down versus bottom-up
11
1.3.4
LGCA versus molecular dynamics
11
2.
Cellular Automata
15
2.1
What are cellular automata?
15
2.2
A short history of cellular automata
16
2.3
One-dimensional cellular automata
17
2.3.1
Qualitative characterization of one-dimensional cellu-
lar automata .
23
2.4
Two-dimensional cellular automata
29
2.4.1
Neighborhoods in 2D
29
2.4.2
Fredkin's game
30
2.4.3
'Life'
31
2.4.4
CA: what else? Further reading
35
2.4.5
From CA to LGCA
36
VI
Table of Contents
3. Lattice-gas cellular automata
39
3.1 The HPP lattice-gas cellular automata
39
3.1.1 Model description
39
3.1.2 Implementation of the HPP model: How to code
lattice-gas cellular automata?
44
3.1.3 Initialization
48
3.1.4 Coarse graining
50
3.2 The FHP lattice-gas cellular automata
53
3.2.1 The lattice and the collision rules
53
3.2.2 Microdynamics of the FHP model
59
3.2.3 The Liouville equation
64
3.2.4 Mass and momentum density
65
3.2.5 Equilibrium mean occupation numbers
66
3.2.6 Derivation of the macroscopic equations: multi-scale
analysis
3.2.7 Boundary conditions
3.2.8 Inclusion of body forces
3.2.9 Numerical experiments with FHP
3.2.10 The 8-bit FHP model
3.3 Lattice tensors and isotropy in the macroscopic limit
Lattice tensors: single-speed models
Generalized lattice tensors for multi-speed models
95
Thermal LBMs: D2Q13-FHP (multi-speed FHP mode!) 101
Exercises
104
3.4 Desperately seeking a lattice for simulations in three dimen-
sions
105
3.4.1 Three dimensions
105
3.4.2 Five and higher dimensions
108
3.4.3 Four dimensions
109
3.5 FCHC
113
3.5.1 Isometric collision rules for FCHC by Henon
113
3.5.2 FCHC, computers and modified collision rules
114
3.5.3 Isometric rules for HPP and FHP
115
3.3.1
3.3.2
3.3.3
3.3.4
3.3.5
Isotropic tensors
69
79
80
83
87
90
90
91
Table of Contents
VII
3.5.4
What else?
116
3.6
The pair interaction (PI) lattice-gas cellular automata
118
3.6.1
Lattice, cells, and interaction in 2D
118
3.6.2
Macroscopic equations
121
3.6.3
Comparison of PI with FHP and FCHC
124
3.6.4
The collision operator and propagation in C and FOR-
TRAN
124
3.7
Multi-speed and thermal lattice-gas cellular automata
128
3.7.1
The D3Q19 model
128
3.7.2
The D2Q9 model
131
3.7..3
The D2Q21 model
134
3.7.4
Transsonic and supersonic fiows: D2Q25, D2Q57,
D2Q129
134
3.8
Zanetti (`staggered') invariants
135
3.8.1
FHP
135
3.8.2
Significance of the Zanetti invariants
135
3.9
Lattice-gas cellular automata: What else?
137
4.
Some statistical mechanics
139
4.1
The Boltzmann equation
139
4.1.1
Five collision invariants and Maxwell's distribution
140
4.1.2
Boltzmann's H-theorem
141
4.1.3
The BGK approximation
143
4.2
Chapman-Enskog: From Boltzmann to Navier-Stokes
145
4.2.1
The conservation laws
146
4.2.2
The Euler equation
147
4.2.3
Chapman-Enskog expansion
147
4.3
The maximum entropy principle
153
5.
Lattice Boltzmann Models
159
5.1
From lattice-gas cellular automata to lattice Boltzmann mod-
els
159
5.1.1
Lattice Boltzmann equation and Boltzmann equation
160
5.1.2
Lattice Boltzmann models of the first generation
163
5.2
BGK lattice Boltzmann model in 2D
165
Table of Contents
5.2.1 Derivation of the W
i
170
5.2.2 Entropy and equilibrium distributions
171
5.2.3 Derivation of the Navier-Stokes equations by multi-
scale analysis
174
5.2.4 Storage demand
182
5.2.5 Simulation of two-dimensional decaying turbulence
183
5.2.6 Boundary conditions for LBM
189
5.3 Hydrodynamic lattice Boltzmann models in 3D
195
5.3.1 3D-LBM with 19 velocities
195
5.3.2 3D-LBM with 15 velocities and Koelman distribution
196
5.3.3 3D-LBM with 15 velocities proposed by Chen et al.
(D3Q15)
197
5.4 Equilibrium distributions: the ansatz method
198
5.4.1 Multi-scale analysis
199
5.4.2 Negative distribution functions at high speed of sound 203
5.5 Hydrodynamic LBM with energy equation
205
5.6 Stability of lattice Boltzmann models
208
5.6.1 Nonlinear stability analysis of uniform flows
208
5.6.2 The method of linear stability analysis (von Neumann) 210
5.6.3 Linear stability analysis of BGK lattice Boltzmann
models
212
5.6.4 Summary.
215
5.7 Simulating ocean circulation with LBM
219
5.7.1 Introduction
219
5.7.2 The model of Munk (1950)
219
5.7.3 The lattice Boltzmann model
222
5.8 A lattice Boltzmann equation for diffusion
232
5.8.1 Finite differences approximation
232
5.8.2 The lattice Boltzmann model for diffusion
233
5,8.3 Multi-scale expansion
234
5.8.4
The special case w —
1
236
5.8.5 The general case
236
5.8.6 Numerical experiments
236
5.8.7 Summary and conclusion
237