1. Lattice Boltzmann: Introduction
Sauro Succi

2. Relaxation Methods
Let us consider the usual Liouville evolution problem:
Let us cast it in relaxation form :
Where tau is an adjustable relaxation parameter, and the local equilibrium reads:
The map phi to phi^leq is non-local (via L) and possibly non-linear as well.
Note that the local equil is still a function of space and time, but usually
slower than phi itself.

3. “Exact” time marching Enslaving Principle
Time marching: Exact small-time Relaxation Propagator:
Where:
t+h t
This reduces to Euler in the limit h/tau<<1, but may operate beyond this limit
because 0<p<1. Hence in principle with larger timesteps than CFL.
Enslaving Principle
The solution at t+h “forgets” the actual value at time
t on a timescale tau (memory) and on the same time scales it is attracted
to the local equilibrium value. Note that the local equilibrium is a
bootstrap moving target, as it depends on phi(t)

4. Long-time solution: recursion
Compact bi-diagonal recursion (can lead to exact solutions):
2. Long-time solution: nicely resummed zig-zag series
The sum of the coeffs
is (p+q)^n = 1

5. Time asymptotics Steady-State solution:
Thus, the time-asymptotic equilibrium obeys:
(Classical ground state)
This is the time asymptotic equilibrium, to which the local equilibrium is
Converging in the limit t to infinity. The actual solution, being enslaved to the
local equilibrium, also converges to the steady-state
In other words the non-equilibrium component, with lifetime tau, decays to zero:

6. Diffusion in Relaxation form
Looks fully local, but non-locality is hidden in the equilibrium:
The actual solution phi relaxes to phi^leq (ATTRACTOR)
on a time-scale tau (Enslaving).
The attractor changes in space and time, but usually more slowly than phi. The time-asymptotic value delivers steady state:

7. Standard explicit in relax form
The standard Euler Centered scheme:
Takes the relaxation form:
The equil is just the space average…:
It’s a slow mover
Realizability implies:
Namely the standard CFL:
Back to square 1!

8. General Relaxation form
The general local equilibrium for the diffusion equation is:
Which is generally different from
unless tau=h
Thus the relaxation-propagator scheme is more general:
Standard Euler
Centered recovered
The realizability implies:
Which is always true for any h because p is an exponential!
Of course we still worry about accuracy, but stability is no concern
as long as epsilon<1.

9. Relaxation algorithm What do we gain?
1. Compute the self-consistent local equilibrium (slow-mode):
2. Relax around it
What do we gain?
The timestep can go beyond CFL with no stability concern as long
as epsilon<1. Our responsibility to watch accuracy.
2. Faster artificial dynamics to steady state (steady-state accelerator)
3. Spawns remarkable generalizations (in extra-dimensions) of which
Lattice Boltzmann is a special and remarkable case

10. Entry Lattice Boltzmann

11. LB for Diffusion equation in d=1
But we shall generalize to
Conservative and Non-Conservative PDE’s .

12. D1Q2: 2 discrete velocities, d=1
Extra-dimensions: Double degrees of freedom:
Left mover f-
Right mover f+ -c +c

13. Rearrange Double degrees of freedom: Left mover f- Right mover f+ v=-c

14. Taylor expand streaming
Double degrees of freedom:
Left mover f-
Right mover f+ -c =c

15. From pdf to Fields: sum and subtract
Double degrees of freedom:
Left mover f-
Right mover f+ -c +c

16. Identify the fields Define macro-fields: Density, Current, Pressure:
Field Evolution:
Equation of state: csound = c

17. Enslaving to local equil
Define hydrofields: Density, Current, Pressure:
Enslaving:
The current fast-relaxes to Jeq
Insert back:

18. Pin down local equil Enslaving: By requiring:
Which is teh desired Diffusion Equation

19. Solve local equil Mass Conservation Momentum Non-Conservation
We obtain the specific expression of local equil:
Plausible: since nothing propgates, the local equil
must be neutral to left/right motion

20. LB scheme for 1d diffusion
Stream and Collide=Relaxation Dynamics:

21. LB: two-lane communication
Gain:
Streaming is first order in space and time: No second order derivatives: Better CFL
Equilibrium is Local: No derivatives: Better parallel computing
Streaming is EXACT, Collisions are roundoff conservative
Price
Double degrees of freedom
Enslaving approximation

22. Why call it Boltzmann? Boltzmann Equation for actual molecules:
The LHS is the particle free motion, the RHS describes particle collisions.
The latter are generally very complicated, but close to equilibrium can be
written in the much simpler relaxation form omega*(f-f^leq).
The Lattice BE stems from the (extreme) discretization of velocity space:

23. Generalizations
No enslaving+ MassMom Conservation = wave equation in free space
Enslaving with imposed velocity U: advection-diffusion
Three states D1Q3: rho,J,P 1d Navier-Stokes fluids
D>1: Rotational symmetry, next lecture

24. Wave equation
Take d/dt *eq1 – d/dx*eq2 and sum up, no enslaving assumption:
Since we have only two movers:
Hence:
Next impose Mass and Momentum Conservation
Free-wave equation!

25. Local equil for wave equation
Compute the actual current density:
This defines the velocity u
Next impose Mass and Momentum conservation
This delivers:
Since there is propagation, the equilibria
are no longer symmetric; similar to random walk
except that nothing is random here!
Note that u is an intermediate, it does not appear
in the final field equation!

26. Advection-Diffusion Same procedure as before leads to:
Mass Conservation
Momentum Non-Conservation
This delivers:
With enslaving, this yields:

27. Fluid dynamcs d=1 Macroscopic Target: hydrodynamic eqs in d=1
(Isothermal T=const)
With two movers p=rho*c^2, frozen.
If we want freedom on the sound speed a third
PDF is necessary: this leads to D1Q3

28. Fluid dynamics: D1Q3 Mass conserved Mom conserved
MomFlux NON conserved (dissipation)
v=-c
v=0
Rest particle
(active spectator)
v=+c

29. LBE: Fluid dynamics
D1Q3 scheme:
v=-c
v=0
v=+c

30. D1Q3: kinetic moments
A formal projection on {1,c,c^2} delivers a sequence of PDEs:

31. Moment hierarchy Kinetic moment of order n:
The hierarchy is not closed because of non-equilibrium:

32. D1Q3: enslaving Hydrodynamic eqs in d=1
Enslaving + P close to equilibrium
Insert this back into the equation for the current
By comparing with NSE, we recover d=1 hydrodynamics iff:

33. D1Q3: local equilibria Mass-Momentum Conservation
3 fields require (at least) 3 movers
Momentum-Flux Not Conserved!
c*(2)+(3):

34. Fluid dynamics The 3x3 system delivers the local equilibria: Where:
Equation of state, sound/light speed
“Relativistic fluid/light speed”
The relativistic flavor is unsurprising because particles move
at the light speed c=dx/dt: Weyl fermions!

35. “Thermodynamic” EoS Mass-momentum at zero flow (u=0)
This naturally defines a set of weights:
They are the lattice analogues of Boltzmann weights
These are just “analogues” because there is no continuum
distribution in velocity space. No standard thermodynamics!

36. Equils: athermal fluid
T=cs^2/c^2=0 Condensation
T=cs^2/c^2=1 Bimodal Depletion

37. Equils: athermal fluid
T=cs^2/c^2=1/3 “Maxwell-Boltzmann”
This is the LB equation of State= Relativistic Radiation!
Intriguing: this is the mass matrix of hat Finite Elements!

38. LBE: Upsides What did we gain? First order in space and time
Equilibria are local: round-off precision, no space-derivatives!
Streaming is along lightcones, not the material lines: EXACT
Negative numerical diffusion (higher order terms)
Small dispersion
Complex geometries (visible in d>1)
Parallel computing (visible in d>1)

39. LBE: downsides What did we loose?
Holds only under enslaving and closeness to local equilibrium
More memory
Uniform lattices
Fixed time-step
The upsides become much more compelling
in d>1: next lecture!

40. Assignements Write a 1d LB for advection-diffusion
and compare with explicit finite-differences
2. Same for 1d fluid dynamics
3. Same for Advection-Diffusion Reaction: how would you
account for reactions?

41. End of Lecture

42. Relaxation to Equilibrium
Eigenvalues/functions:
Time asymptotic equilibrium: Null space of L: conserved quantities

43. Diffusion in Relaxation form
In discrete form (Euler):
Looks fully local, but non-locality is hidden in the equilibrium:

44. Diffusion = Non-local equil
Equil = Steady state
Local Equil = Space average
Q: can we turn the non-local equil into a LOCAL one?

45. D1Q3: indep pressure (EoS)
Note Momentum Non-Conservation
3 fields require (at least) 3 movers
v=-c
v=-0
v=+c

46. Equils in compact form