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

Source abstract

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 field and a road in a population dynamics context) and coupled through exchange terms on the road, which makes its analysis quite involved. We propose a two-point flux approximation (TPFA) finite volume scheme. In both the continuous and the discrete settings, we prove the dissipation of a quadratic entropy, with some entropy dissipation-entropy relation, leading to the exponential decay in time of the solution towards the stationary state selected by the total mass of the initial data. To deal with the problem of different dimensions in the proof of the entropy dissipation-entropy relation, we artificially “thicken” the road and adapt the proof of the Poincaré–Wirtinger inequality. Numerical simulations confirm and complete the analysis, and raise new issues.