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