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

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

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 3

4. Scheme of LBM approach Microscopic 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 4

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

6. Elementary advection 6

7. Advection 7 Microvariables – f i

8. Macrovariables: 8

9. Properties of collisions 9

10. Equilibria 10

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

12. Invariance equation 12

13. Solution to Invariance Equation 13

14. LBM up to the kth order 14

15. Stability theorem: conditions 15 Contraction is uniform:

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

17. Macroscopic Equations 17

18. Construction of macroscopic equations and matching condition 18

19. 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 20 c~1,u≤Ma

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

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

23. References 23 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]arXiv:1006.3270v1

24. Questions please 24 Vorticity, Re=5000