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arXiv · 2004.13981

The Scharfetter--Gummel scheme for aggregation-diffusion equations

André Schlichting·Christian Seis
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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.

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André Schlichting, Christian Seis. 2020-04-29. The Scharfetter--Gummel scheme for aggregation-diffusion equations. https://doi.org/10.1093/imanum%2Fdrab039

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