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

TL;DR

This paper demonstrates the convergence of the Scharfetter-Gummel scheme for aggregation-diffusion equations using a variational approach, unifying discrete and continuum limits and recovering the gradient flow structure.

Contribution

It introduces a novel gradient structure for finite volume discretizations that applies for any non-negative diffusion constant, enabling simultaneous analysis of discrete-to-continuum and zero-diffusion limits.

Findings

Established convergence in gradient structures for the scheme

Recovered the Otto gradient flow structure in the limit

Unified analysis of discrete and continuum limits

Abstract

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.