Finite volumes for the Stefan-Maxwell cross-diffusion system

TL;DR

This paper introduces a finite volume scheme for the Stefan-Maxwell cross-diffusion system that guarantees convergence and preserves key physical properties like non-negativity, mass conservation, and entropy dissipation.

Contribution

It proposes a novel, provably convergent finite volume scheme that maintains essential physical and mathematical properties of the continuous Stefan-Maxwell model.

Findings

The scheme preserves non-negativity of solutions.

It conserves mass and volume-filling constraints.

Numerical results demonstrate the scheme's effectiveness.

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.