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LBM: Approximate Invariant Manifolds and Stability

LBM: Approximate Invariant Manifolds and Stability. Alexander Gorban (Leicester) Tuesday 07 September 2010, 16:50-17:30 Seminar Room 1, Newton Institute. 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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LBM: Approximate Invariant Manifolds and Stability

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  1. LBM: Approximate Invariant Manifolds and Stability Alexander Gorban (Leicester) Tuesday 07 September 2010, 16:50-17:30 Seminar Room 1, Newton Institute

  2. In LBM “Nonlinearity is local, non-locality is linear”(Sauro Succi) Moreover, in LBM non-locality is linear, exact and explicit

  3. 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

  4. 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

  5. Simplified scheme of LBM Time step expansion for IM “Macroscopic” model (Navier-Stokes) after initial layer Dynamics of discrete lattice Boltzmann model

  6. Elementary advection

  7. Advection Microvariables – fi

  8. Macrovariables:

  9. Properties of collisions

  10. Equilibria

  11. LBM chain f→advection(f) → collision(advection(f))→ advection(collision(advection(f) )) → collision(advection(collision(advection(f))) →...

  12. Invariance equation

  13. Solution to Invariance Equation

  14. LBM up to the kth order

  15. Stability theorem:conditions Contraction is uniform:

  16. Stability theorem There exist such constants That for The distance from f(t) to the kth order invariant manifold is less than Cεk+1

  17. Macroscopic Equations

  18. Construction of macroscopic equations and matching condition

  19. Space discretization: if the grid is advection-invariant then no efforts are needed 19

  20. 1D athermal equilibrium, v={0,±1}, T=1/3, matching moments, BGK collisions c~1,u≤Ma

  21. 2D Athermal 9 velocities model (D2Q9), equilibrium

  22. 2D Athermal 9 velocities model (D2Q9) c~1,u≤Ma

  23. 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]

  24. Questions please Vorticity, Re=5000

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