Finite‐volume scheme for a degenerate cross‐diffusion model motivated from ion transport

Abstract

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.

Convergence of a positive nonlinear control volume finite element scheme for an anisotropic seawater intrusion model with sharp interfaces

“…In , the authors studied the large time behavior for the aforementioned model and exhibited its numerical convergence rate toward a steady state. For a degenerate cross‐diffusion model describing the ion transport through biological membranes, a finite volume scheme is analyzed in .…”

Comparison of a finite-element and finite-volume scheme for a degenerate cross-diffusion system for ion transport

“…A lattice-free approach, starting from stochastic Langevin equations, can be found in [2]. The scope of this paper is to present a new finite-element discretization of the degenerate cross-diffusion system and to compare this scheme to a previously proposed finite-volume method [5].…”

“…It is reflected in the entropy inequality since we lose the gradient estimate if u i = 0 or u 0 = 0. This problem is overcome by using the "degenerate" Aubin-Lions lemma of [21,Appendix C] or its discrete version in [5,Lemma 10].…”

“…We are interested in devising a numerical scheme which preserves the structure of the continuous system, like nonnegativity, upper bounds, and the entropy structure, on the discrete level. A first result in this direction was presented in [5], analyzing a finitevolume scheme preserving the aforementioned properties. However, the scheme preserves the nonnegativity and upper bounds only if the diffusion coefficients D i are all equal, and the discrete entropy is dissipated only if additionally the potential term vanishes.…”

“…Before we proceed, we briefly discuss some related literature. While there are many results for the classical Poisson-Nernst-Planck system, see for example [25,29], there seems to be no numerical analysis of the ion-transport model (1)-(4) apart from the finitevolume scheme in [5] and simulations of the stationary equations in [4]. Let us mention some other works on finite-element methods for related cross-diffusion models.…”