The Scharfetter–Gummel scheme for aggregation-diffusion equations

04/29/2020

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by   André Schlichting, et al.

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In this paper, we propose a finite-volume scheme for aggregation-diffusion equations that is based on a Scharfetter–Gummel approximation of the nonlinear, 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, and, as a consequence, the numerical solutions converge in the large time limit to stationary solutions, for which we provide a thermodynamic characterization.