LBM: Approximate Invariant Manifolds and Stability
Alexander Gorban (Leicester)Tuesday 07 September 2010,
16:50-17:30Seminar 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-Macrovariables-
Collisions- 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 approachMicroscopic model(The Boltzmann Equation)
Asymptotic Expansion
“Macroscopic” model (Navier-Stokes)
Discretization in velocity space
Finite velocity model
Discretization in space and time
Discrete lattice Boltzmann model
Approximation
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Simplified scheme of LBM
“Macroscopic” model (Navier-Stokes)after initial layer
Dynamics of discrete lattice Boltzmann model
Time step expansion for IM
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Elementary advection
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Advection
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Microvariables – fi
Macrovariables:
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Properties of collisions
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Equilibria
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LBM chain
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f→advection(f) → collision(advection(f))→ advection(collision(advection(f) )) → collision(advection(collision(advection(f))) →...
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
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),...,1,0(sup
),...,1,0(sup
),...,1,0(sup
1
eq1
1
kjBD
kjAfD
kjAfD
jM
jMx
jjxx
jjxx
Contraction is uniform: 1M
Stability theorem
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There exist such constants
),...,,,...,(),,...,,,...,( 111111111 kkkk BBAACBBAAC
That for 10suplnln
ln
1Cfk
t
The distance from f(t) to the kth order invariant manifold is less than Cεk+1
1
0)( ),(),(
kk
j
kfm Cxtfxtf
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
20c~1,u≤Ma
2D Athermal 9 velocities model (D2Q9), equilibrium
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2D Athermal 9 velocities model (D2Q9)
22c~1,u≤Ma
References
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•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]
Questions please
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Vorti
city
, Re=
5000