The Scharfetter--Gummel scheme for aggregation-diffusion equations

Abstract

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.

37 pages. Accepted version with existence and stability argument without CFL-conditions. In addition, some numerical experiments are added

References in corpus (2)

• Jump processes as Generalized Gradient Flows
• An invariance principle for gradient flows in the space of probability measures

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