The Scharfetter-Gummel scheme for aggregation-diffusion equations

In this paper we propose a finite-volume scheme for aggregation-diffusion equations based on a Scharfetter-Gummel approximation of the quadratic, nonlocal flux term. This scheme is analyzed concerning well posedness and convergence towards solutions to the continuous problem. Also, it is proven that the numerical scheme has several structure-preserving features. More specifically, it is shown that the discrete solutions satisfy a free-energy dissipation relation analogous to the continuous model. Consequently, the numerical solutions converge in the large time limit to stationary solutions, for which we provide a thermodynamic characterization. Numerical experiments complement the study.