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LBM: Approximate Invariant Manifolds and Stability

Explore the concepts of invariant manifolds and stability in lattice Boltzmann models, covering definitions, solutions, stability theorems, and applications to macroscopic equations. Discover the intricate dynamics of discrete lattice Boltzmann models and their relation to the Boltzmann equation. References to prominent works such as Succi's book on lattice Boltzmann equations and Gorban's work on invariant manifolds offer valuable insights into this topic. Join the seminar facilitated by Alexander Gorban to delve deeper into these fascinating concepts in LBM.

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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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