An implicit Euler finite‐volume scheme for a degenerate cross‐diffusion system describing the ion transport through biological membranes is proposed. 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 degeneracies, which lead to considerable mathematical difficulties. The finite‐volume scheme is based on two‐point flux approximations with “double” upwind mobilities. The existence of solutions to the fully discrete scheme is proved. When the particles are not distinguishable and the dynamics is driven by cross diffusion only, it is shown that the scheme preserves the structure of the equations like nonnegativity, upper bounds, and entropy dissipation. The degeneracy is overcome by proving a new discrete Aubin–Lions lemma of “degenerate” type. Numerical simulations of a calcium‐selective ion channel in two space dimensions show that the scheme is efficient even in the general case of ion transport.

中文翻译：

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由离子输运驱动的简并交叉扩散模型的有限体积方案

提出了一种简并交叉扩散系统的隐式欧拉有限体积方案，描述了离子通过生物膜的传输。离子浓度的强耦合方程包括涉及电势的漂移项，电势通过泊松方程与浓度耦合。交叉扩散系统具有揭示非标准简并性的形式梯度流结构，这会导致相当大的数学困难。有限体积方案基于具有“双”逆风流动性的两点通量近似。证明了全离散格式解的存在性。当粒子不可区分并且动力学仅由交叉扩散驱动时，结果表明该方案保留了方程的结构，如非负性、上限和熵耗散。通过证明“简并”类型的新离散 Aubin-Lions 引理克服了简并性。二维空间中钙选择性离子通道的数值模拟表明，即使在一般的离子传输情况下，该方案也是有效的。