Multiscale analysis for convection dominated transport equations

Pages: 281 - 298, Volume 23, Issue 1/2, January/February 2009

doi:10.3934/dcds.2009.23.281       Abstract        Full Text (246.8K)

Thomas Y. Hou - Department of Applied and Computational Mathematics, California Institute of Technology, Pasadena, CA 91125, United States (email)
Dong Liang - Department of Mathematics and Statistics, York University, Toronto, ON, M3J 1P3, Canada (email)

Abstract: In this paper, we perform a systematic multiscale analysis for convection dominated transport equations with a weak diffusion and a highly oscillatory velocity field. The paper primarily focuses on upscaling linear transport equations. But we also discuss briefly how to upscale two-phase miscible flows, in which case the concentration equation is coupled to the pressure equation in a nonlinear fashion. For the problem we consider here, the local Peclet number is of order $O(\epsilon^{-m+1})$ with $m \in [2,\infty]$ being any integer, where $\epsilon$ characterizes the small scale in the heterogeneous media. Due to the presence of the nonlocal memory effect, upscaling a convection dominated transport equation is known to be very difficult. One of the key ideas in deriving a well-posed homogenized equation for the convection dominated transport equation is to introduce a projection operator which projects the fluctuation onto a suitable subspace. This projection operator corresponds to averaging along the streamlines of the flow. In the case of linear convection dominated transport equations, we prove the well-posedness of the homogenized equations and establish rigorous error estimates for our multiscale expansion.

Keywords:  Multiscale analysis, nonlocal memory effect, error analysis, heterogeneous porous media, two-phase miscible flow
Mathematics Subject Classification:  Primary: 65N30, 74Q15; Secondary: 76M50

Received: February 2008;      Revised: July 2008;      Published: September 2008.