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.
Keywords
Cite
@article{arxiv.2004.13981,
title = {The Scharfetter--Gummel scheme for aggregation-diffusion equations},
author = {André Schlichting and Christian Seis},
journal= {arXiv preprint arXiv:2004.13981},
year = {2021}
}
Comments
37 pages. Accepted version with existence and stability argument without CFL-conditions. In addition, some numerical experiments are added