Numerical Studies of Homogenization under a Fast Cellular Flow

Abstract

Abstract. We consider a two dimensional particle diffusing in the presence of a fast cellular flow confined to a finite domain. If the flow amplitude A is held fixed, and the number of cells L 2 → ∞, then problem homogenizes, and has been well studied. Also well studied is the limit when L is fixed, and A → ∞. In this case the solution averages along stream lines. The double limit as both the flow amplitude A → ∞ and the number of cells L 2 → ∞ was recently studied [9], one observes a sharp transition between the homogenization and averaging regimes occurring at A ≈ L 4 . This paper numerically studies a few theoretically unresolved aspects of this problem when both A and L are large that were left open in [9] using the numerical method devised in [16]. Our treatment of the numerical method uses recent developments in the theory of modified equations for numerical integrators of SDEs [23].

From averaging to homogenization in cellular flows – An exact description of the transition

“…The methods in [IKNR13] are analytic, based on investigating the asymptotic behavior of the principal Dirichlet eigenvalue of the elliptic operator, and it seems unlikely that they can be directly applied near the transition regime. To our knowledge, only numerical results were available in the intermediate cases [IZ12,PSZ09] up until now.…”