  In this paper, we consider a class of convection-diffusion equations with
memory effects. These equations arise as a result of homogenization or
upscaling of linear transport equations in heterogeneous media and play an
important role in many applications. First, we present a dememorization
technique for these equations. We show that the convection-diffusion equations
with memory effects can be written as a system of standard
convection-diffusion-reaction equations. This allows removing the memory term
and simplifying the computations. We consider a relation between dememorized
equations and micro-scale equations, which do not contain memory terms. We note
that dememorized equations differ from micro-scale equations and constitute a
macroscopic model. Next, we consider both implicit and partially explicit
methods. The latter is introduced for problems in multiscale media with
high-contrast properties. Because of high contrast, explicit methods are
restrictive and require time steps that are very small (scales as the inverse
of the contrast). We show that, by appropriately decomposing the space, we can
treat only a few degrees of freedom implicitly and the remaining degrees of
freedom explicitly. We present a stability analysis. Numerical results are
presented that confirm our theoretical findings of partially explicit schemes
applied to dememorized systems of equations.
