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.

arXiv comment37 pages. Accepted version with existence and stability argument without CFL-conditions. In addition, some numerical experiments are added

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Cited by in corpus (7)

• Cosh gradient systems and tilting
• Variational convergence of the Scharfetter-Gummel scheme to the aggregation-diffusion equation and vanishing diffusion limit
• On a Class of Nonlocal Continuity Equations on Graphs
• Homogenization of Wasserstein gradient flows
• Error estimates for a finite volume scheme for advection-diffusion equations with rough coefficients
• Coarse-graining nonequilibrium diffusions with Markov chains
• Variational Asymptotic Preserving Scheme for the Vlasov-Poisson-Fokker-Planck System