Variational convergence of the Scharfetter-Gummel scheme to the aggregation-diffusion equation and vanishing diffusion limit

In this paper, we explore the convergence of the semi-discrete Scharfetter-Gummel scheme for the aggregation-diffusion equation using a variational approach. Our investigation involves obtaining a novel gradient structure for the finite volume space discretization that works consistently for any non-negative diffusion constant. This allows us to study the discrete-to-continuum and zero-diffusion limits simultaneously. The zero-diffusion limit for the Scharfetter-Gummel scheme corresponds to the upwind finite volume scheme for the aggregation equation. In both cases, we establish a convergence result in terms of gradient structures, recovering the Otto gradient flow structure for the aggregation-diffusion equation based on the 2-Wasserstein distance.