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

About the entropic structure of detailed balanced multi-species cross-diffusion equations

“…We also assume continuous positive initial data u 0 i := u 0 i (x) > 0 on T for all 1 ≤ i ≤ n. Then for all M ∈ N, there exists a unique global solution u i := u i (t, x k ) > 0 of class C ∞ to the discrete system (9) with h := 1/M. Denoting [ũ i ] M the interpolant obtained from u i (t, x k ) by formula (10), then there exists a subsequence such that the following holds:…”

“…With the help of the following lemma, we can switch between the discrete norm and the norm of the continuous linear interpolant w of w defined in (10):…”

“…For the second rigorous limit from the space discretised master equation to the SKT cross-diffusion system, similar discrete in space approximation schemes for the SKT model were studied in [7,8,9], but they are not necessarily entropy preserving. Very recently, an entropy preserving numerical scheme was proposed in [10], though not for the SKT model, but for a volume-filling type cross-diffusion system.…”

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].…”

“…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.…”