Long time behavior of the field-road diffusion model: an entropy method and a finite volume scheme

TL;DR

This paper analyzes the long-term behavior of a coupled field-road diffusion model using an entropy method and proposes a finite volume scheme, demonstrating exponential decay to equilibrium and confirming results through numerical simulations.

Contribution

It introduces a novel finite volume scheme for the field-road diffusion model and proves exponential decay of entropy in both continuous and discrete settings.

Findings

Proves exponential decay of entropy and convergence to stationary state.

Develops a TPFA finite volume scheme for the model.

Numerical simulations support theoretical results.

Abstract

We consider the so-called field-road diffusion model in a bounded domain, consisting of two parabolic PDEs posed on sets of different dimensions (a {\it field} and a {\it road} in a population dynamics context) and coupled through exchange terms on the road, which makes its analysis quite involved. We propose a TPFA finite volume scheme. In both the continuous and the discrete settings, we prove theexponential decay of an entropy, and thus the long time convergence to the stationary state selected by the total mass of the initial data. To deal with the problem of different dimensions, we artificially \lq\lq thicken'' the road and, then, establish a rather unconventional Poincar{\'e}-Wirtinger inequality. Numerical simulations confirm and complete the analysis, and raise new issues.