On the upscaling of the diffusion equation in a heterogeneous medium by the two different methods

Upscaling the flow equations appears in many studies related to diffusion, heat conductivity, and flows in porous media. Variable coefficients of the original fine-scale process description are substituted by averaged constant values. The different upscaling procedures have been suggested in the literature. The question arises, whether they result in the same or different upscaled models; moreover, whether the solutions of the coarse-scale equations provide a reasonably accurate description of the fine scale. In this work, we consider three sample 1D diffusion problems with periodic coefficients: diffusion with or without the external source, as well as diffusion with advection. These problems are upscaled by the two methods. The method of direct upscaling selects the averaged coefficients to provide the minimum difference between the solutions of the coarse-scale and fine-scale models. The method of continuous upscaling, developed previously, consists of the continuous averaging transformation between the distant scales. New expressions for the upscaled diffusion coefficients were derived for this case. It turns out that the direct upscaling results in multiple optimum parameters of the upscaled model. Meanwhile, continuous upscaling points at one of them. The coarse-scale approximation of a fine-scale solution may be unsatisfactory, even for the best choice of the upscaled coefficient. A numerical study demonstrates a nontrivial behavior of the diffusion coefficient under continuous upscaling, while it converges to an asymptotic value.