LBM: Approximate Invariant Manifolds and Stability
Seminar Room 1, Newton Institute
We study the Lattice Boltzmann Models in the framework of the Geometric Singular Perturbation theory. We begin with the Lattice Boltzmann system discrete in both velocity space and time with the alternating steps of advection and relaxation, common to all lattice Boltzmann schemes. When time step is small then this system has an approximate invariant manifold close to locally equilibrium distributions. We found a time step expansion for the approximate invariant manifold and proved its conditional stability in any order of accuracy under condition that the space derivatives of the correspondent order remain bounded. On this invariant manifold, a macroscopic dynamics arises and we found the time step expansion of the equation of the macroscopic dynamics.