American Institute of Mathematical Sciences

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January & February  2009, 23(1&2): 281-298. doi: 10.3934/dcds.2009.23.281

Multiscale analysis for convection dominated transport equations

Thomas Y. Hou 1, and  Dong Liang 2,

1. 

Department of Applied and Computational Mathematics, California Institute of Technology, Pasadena, CA 91125, United States
2. 

Department of Mathematics and Statistics, York University, Toronto, ON, M3J 1P3, Canada

Received  February 2008 Revised  July 2008 Published  September 2008

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: nonlocal memory effect, two-phase miscible flow, Multiscale analysis, error analysis, heterogeneous porous media.
Mathematics Subject Classification: Primary: 65N30, 74Q15; Secondary: 76M50.
Citation: Thomas Y. Hou, Dong Liang. Multiscale analysis for convection dominated transport equations. Discrete & Continuous Dynamical Systems - A, 2009, 23 (1&2) : 281-298. doi: 10.3934/dcds.2009.23.281
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