Nonlocal transport equations in multiscale media. Modeling, dememorization, and discretizations

TL;DR

This paper introduces a dememorization technique for convection-diffusion equations with memory effects, simplifying computations and enabling efficient multiscale simulations with high-contrast media using partially explicit methods.

Contribution

The paper develops a novel dememorization approach transforming memory equations into standard convection-diffusion-reaction systems, facilitating efficient numerical solutions in multiscale heterogeneous media.

Findings

Dememorization simplifies equations by removing memory terms.

Partially explicit methods improve computational efficiency in high-contrast media.

Numerical results confirm stability and effectiveness of proposed schemes.

Abstract

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.