LBM: Approximate Invariant
Manifolds and Stability
Alexander Gorban (Leicester)
Tuesday 07 September 2010,
16:50-17:30
Seminar Room 1, Newton Institute
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In LBM
“Nonlinearity is local,
non-locality is linear”
(Sauro Succi)
Moreover, in LBM
non-locality is linear,
exact and explicit
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Plan
• Two ways for LBM definition
• Building blocks: Advection-MacrovariablesCollisions- Equilibria
• Invariant manifolds for LBM chain and Invariance
Equation,
• Solutions to Invariance Equation by time step
expansion, stability theorem
• Macroscopic equations and matching conditions
• Examples
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Scheme of LBM approach
Microscopic model
(The Boltzmann Equation)
Discretization in
velocity space
Asymptotic
Expansion
Finite velocity
model
“Macroscopic” model
(Navier-Stokes)
Discretization in
space and time
Approximation
Discrete lattice
Boltzmann model
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Simplified scheme of LBM
Dynamics of discrete
lattice Boltzmann
model
Time step
expansion for IM
“Macroscopic” model
(Navier-Stokes)
after initial layer
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Elementary advection
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Advection
Microvariables – fi
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Macrovariables:
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Properties of collisions
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Equilibria
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LBM chain
f→advection(f) → collision(advection(f))→
advection(collision(advection(f) )) →
collision(advection(collision(advection(f))) →...
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Invariance equation
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Solution to Invariance Equation
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LBM up to the kth order
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Stability theorem:
conditions
j 1
x
sup x D
j 1
x
sup x D
j 1
M
sup x D
f  A j ( j  0,1,...,k )
f
eq
M
 A j ( j  0,1,...,k )
 B j ( j  0,1,...,k )
Contraction is uniform:
M    1
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Stability theorem
There exist such constants
C( A1,..., Ak 1, B1,...,Bk 1 ), C1 ( A1,..., Ak 1, B1,...,Bk 1 )
That for
1
k ln   ln sup f0  C1 

 ln 
t
The distance from f(t) to the kth order invariant
manifold is less than Cεk+1
k
f (t , x)   f
j 0
k
m( f )
(t , x)  C
k 1
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Macroscopic Equations
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Construction of macroscopic equations
and matching condition
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Space discretization:
if the grid is advection-invariant
then no efforts are needed
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1D athermal equilibrium, v={0,±1}, T=1/3,
matching moments, BGK collisions
c~1,u≤Ma
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2D Athermal 9 velocities model (D2Q9),
equilibrium
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2D Athermal 9 velocities model (D2Q9)
c~1,u≤Ma
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References
•Succi, S.: The lattice Boltzmann equation for fluid dynamics
and beyond. Oxford University Press, New York (2001)
•He, X., Luo., L. S.: Theory of the lattice Boltzmann method:
From the Boltzmann equation to the lattice Boltzmann
Equation. Phys Rev E 56(6) (1997) 6811–6817
•Gorban, A. N., Karlin, I. V.: Invariant Manifolds for Physical
and Chemical Kinetics. Springer, Berlin – Heidelberg (2005)
•Packwood, D.J., Levesley, J., Gorban A.N.: Time Step
Expansions and the Invariant Manifold Approach to Lattice
Boltzmann Models, arXiv:1006.3270v1 [cond-mat.stat-mech]
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Vorticity, Re=5000
Questions please
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