# Advection improves homogenized models of continuum diffusion in one-dimensional heterogeneous media

## TL;DR

This paper introduces an advection term into homogenized diffusion models for one-dimensional heterogeneous media, resulting in more accurate approximations of the original complex system.

## Contribution

It presents a novel method to compute effective advection and diffusion coefficients by solving boundary value problems, enhancing homogenized models of heterogeneous diffusion.

## Key findings

- Including advection improves model accuracy
- Effective coefficients depend on boundary conditions
- Method outperforms standard homogenization approaches

## Abstract

We propose an alternative method for one-dimensional continuum diffusion models with spatially variable (heterogeneous) diffusivity. Our method, which extends recent work on stochastic diffusion, assumes the constant-coefficient homogenized equation takes the form of an advection-diffusion equation with effective (diffusivity and velocity) coefficients. To calculate the effective coefficients, our approach involves solving two uncoupled boundary value problems over the heterogeneous medium and leads to coefficients depending on the spatially-varying diffusivity (as usual) as well as the boundary conditions imposed on the heterogeneous model. Computational experiments comparing our advection-diffusion homogenized model to the standard homogenized model demonstrate that including an advection term in the homogenized equation leads to improved approximations of the solution of the original heterogeneous model.