André Schlichting

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

4 June 2023

Together with Anastasiia Hraivoronska and Oliver Tse, we explore the convergence of the Scharfetter-Gummel scheme for the aggregation-diffusion equation using a variational approach. Our investigation involves obtaining a novel gradient structure for the finite volume scheme that works consistently for any nonnegative diffusion constant, which 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.

The diagram depicts the main results of the paper. The labels on the arrows indicate the corresponding convergence statements in the sense of the EDP convergence. In addition, the generalized gradient structure for the Scharfetter–Gummel scheme is established.
The diagram depicts the main results of the paper. The labels on the arrows indicate the corresponding convergence statements in the sense of the EDP convergence. In addition, the generalized gradient structure for the Scharfetter–Gummel scheme is established.

Now published in Num. Math. 156(6), 2221–2292 (2024).

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