C. Cancès, C. Chainais-Hillairet, A. Gerstenmayer, A. Jüngel:

"Convergence of a finite-volume scheme for a degenerate cross-diffusion model for ion transport";

in: "ASC Report 6/2018", issued by: Institute for Analysis and Scientific Computing; Vienna University of Technology, Wien, 2018, ISBN: 978-3-902627-11-7, 1 - 33.

An implicit Euler ﬁnite-volume scheme for a degenerate cross-diffusion system describing the ion transport through biological membranes is analyzed. The strongly coupled equations for the ion concentrations include drift terms involving the electric potential, which is coupled to the concentrations through the Poisson equation. The cross-diffusion system possesses a formal gradient-ﬂow structure revealing nonstandard degen-eracies, which lead to considerable mathematical diﬃculties. The ﬁnite-volume scheme is based on two-point ﬂux approximations with "double" upwind mobilities. It preserves the structure of the continuous model like nonnegativity, upper bounds, and entropy dissipation. The degeneracy is overcome by proving a new discrete Aubin-Lions lemma of "degenerate" type. Under suitable assumptions, the existence and uniqueness of bounded discrete solutions, a discrete entropy inequality, and the convergence of the scheme is proved. Numerical simulations of a calcium-selective ion channel in two space dimensions indicate that the numerical scheme is of ﬁrst order.

Ion transport, ﬁnite-volume method, gradient ﬂow, entropy method, existence of discrete solutions, convergence of the scheme, calcium-selective ion channel.

http://www.asc.tuwien.ac.at/preprint/2018/asc06x2018.pdf

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