Homogenization for a Parabolic Partial Differential Equation with Two-scaled Advection

Abstract

Advection-diffusion equations are frequently encountered in many fields and have become a very active research area. However, in most cases, these equations concern different time and space scales, which make it impossible to derive explicit solutions to these equations. To overcome this difficulty, the theory of homogenization aims to approximate the original differential equation with rapidly oscillating coefficients by an effective homogenized equation with constant or slowly varying coefficients. The homogenized equation is often quite suitable for theoretical analysis or numerical methods. This paper investigates the homogenization principle of an advection-diffusion partial differential equation. The novelty of the parabolic partial differential equation we consider is that the advection term in the equation is two-scaled, which is rarely considered by others for the homogenization of advection-diffusion equation. Under certain proper assumptions on the coefficient functions of the original advection-diffusion partial differential equation, which ensure the variable elimination, we derive the homogenized equation, which is also an advection-diffusion equation, by the technique of multi-scale expansion. It is shown that the coefficient functions of the original two-scaled equation have different influence on the coefficient functions of the homogenized equation.