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

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

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

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

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

Elementary advection 6

Advection 7 Microvariables – f i

Macrovariables: 8

Properties of collisions 9

Equilibria 10

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

Invariance equation 12

Solution to Invariance Equation 13

LBM up to the kth order 14

Stability theorem: conditions 15 Contraction is uniform:

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

Macroscopic Equations 17

Construction of macroscopic equations and matching condition 18

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

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

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

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

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: v1 [cond-mat.stat-mech]arXiv: v1

Questions please 24 Vorticity, Re=5000