Finite volumes for the Stefan-Maxwell cross-diffusion system

Abstract

The aim of this work is to propose a provably convergent finite volume scheme for the so-called Stefan-Maxwell model, which describes the evolution of the composition of a multi-component mixture and reads as a cross-diffusion system. The scheme proposed here relies on a two-point flux approximation, and preserves at the discrete level some fundamental theoretical properties of the continuous models, namely the non-negativity of the solutions, the conservation of mass and the preservation of the volume-filling constraints. In addition, the scheme satisfies a discrete entropy-entropy dissipation relation, very close to the relation which holds at the continuous level. In this article, we present this scheme together with its numerical analysis, and finally illustrate its behaviour with some numerical results.

Structure Preserving Finite Volume Approximation of Cross-Diffusion Systems Coupled by a Free Interface

“…with density h α (c) = n i=1 c i (log(c i ) − µ * ,α i ) − c i + 1, for α ∈ {s, g}, can be shown to satisfy, for some positive semi-definite mobility matrices M s , M g , the free energy dissipation relation [4,5]…”

A convergent finite volume scheme for dissipation driven models with volume filling constraint

“…In fact there exists only few papers on this subject. Let us cite [23,25,28,40] where the authors studied, using similar techniques than in this paper, some convergent finite volume schemes for an ion transport, the Maxwell-Stefan, a thin-film solar cells and a biofilm cross-diffusion system with a volume filling constraint. However, in these former works the numerical schemes were designed for particular models.…”

“…) be given for each K ∈ T . Then, the vector (u k , π k ) is solution to the following nonlinear system (25) m(K)…”

“…Let us notice that we design our scheme in such a way that u k 0,K also satisfies an equation of the form (25) for every K ∈ T and k ≥ 1. Indeed, summing over i = 1, .…”

“…We proceed with the limit m → ∞ in (25). For this, we set ϕ k K = ϕ(x K , t k ), multiply (25) by ∆t m ϕ k−1 K and sum over K ∈ T m , leading to…”