The Scharfetter–Gummel scheme for aggregation–diffusion equations

Schlichting André, Seis Christian

Abstract

In this paper we propose a finite-volume scheme for aggregation-diffusion equations based on aScharfetter-Gummel approximation of the quadratic, nonlocal flux term. This scheme is analyzedconcerning well posedness and convergence towards solutions to the continuous problem. Also, it isproven that the numerical scheme has several structure-preserving features. More specifically, it is shownthat 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 whichwe provide a thermodynamic characterization. Numerical experiments complement the study.