Volume 55, Number 6, November-December 2021
|Page(s)||3017 - 3042|
|Published online||17 December 2021|
Consistency and convergence for a family of finite volume discretizations of the Fokker–Planck operator
WIAS Berlin, Mohrenstraße 39, 10117 Berlin, Germany
* Corresponding author: firstname.lastname@example.org
Accepted: 17 November 2021
We introduce a family of various finite volume discretization schemes for the Fokker–Planck operator, which are characterized by different Stolarsky weight functions on the edges. This family particularly includes the well-established Scharfetter–Gummel discretization as well as the recently developed square-root approximation (SQRA) scheme. We motivate this family of discretizations both from the numerical and the modeling point of view and provide a uniform consistency and error analysis. Our main results state that the convergence order primarily depends on the quality of the mesh and in second place on the choice of the Stolarsky weights. We show that the Scharfetter–Gummel scheme has the analytically best convergence properties but also that there exists a whole branch of Stolarsky means with the same convergence quality. We show by numerical experiments that for small convection the choice of the optimal representative of the discretization family is highly non-trivial, while for large gradients the Scharfetter–Gummel scheme stands out compared to the others.
Mathematics Subject Classification: 35Q84 / 49M25 / 65N08
Key words: Finite volume / Fokker–Planck / Stolarsky mean
© The authors. Published by EDP Sciences, SMAI 2021
This is an Open Access article distributed under the terms of the Creative Commons Attribution License (https://creativecommons.org/licenses/by/4.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
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