Convergence of a Finite-Volume Scheme for a Degenerate Cross-Diffusion Model for Ion Transport

TL;DR

This paper introduces a finite-volume scheme for a complex degenerate cross-diffusion model of ion transport, proving its convergence and preserving key physical properties, with numerical validation on calcium channels.

Contribution

It develops a novel finite-volume scheme that handles degeneracies and preserves the model's structure, with rigorous convergence and stability analysis.

Findings

The scheme converges to the true solution under certain conditions.

It maintains nonnegativity, upper bounds, and entropy dissipation.

Numerical simulations confirm first-order accuracy on calcium channels.

Abstract

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 dis-sipation. 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.