An entropy structure preserving space-time formulation for cross-diffusion systems: Analysis and Galerkin discretization

Braukhoff, Marcel; Perugia, Ilaria; Stocker, Paul

doi:10.48550/arxiv.2006.13069

Cited by 2 publications

(2 citation statements)

References 43 publications

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“…Closer to our study we mention [26], where a model admitting a gradient flow structure for a constraint Wasserstein metric, similar to the one presented in Section 2.1, is discretized thanks to a minimizing JKO scheme [12,59]. An other approach dealing with discontinuous Galerkin method has been used in [21,72,73]. Finally, we also refer to [2,29,57] for numerical schemes based on the control volume finite element method which preserved the entropy structure of some dissipative models.…”

Section: 3mentioning

confidence: 99%

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

Cancès

Zurek

2022

Numer. Math.

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In this paper we propose and study an implicit finite volume scheme for a general model which describes the evolution of the composition of a multi-component mixture in a bounded domain. We assume that the whole domain is occupied by the different phases of the mixture which leads to a volume filling constraint. In the continuous model this constraint yields the introduction of a pressure, which should be thought as a Lagrange multiplier for the volume filling constraint. The pressure solves an elliptic equation, to be coupled with parabolic equations, possibly including cross-diffusion terms, which govern the evolution of the mixture composition. Besides the system admits an entropy structure which is at the cornerstone of our analysis. More precisely, the main objective of this work is to design a two-point flux approximation finite volume scheme which preserves the key properties of the continuous model, namely the volume filling constraint and the control of the entropy production. Thanks to these properties, and in particular the discrete entropy-entropy dissipation relation, we are able to prove the existence of solutions to the scheme and its convergence. Finally, we illustrate the behavior of our scheme through different applications.

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Section: 3mentioning

confidence: 99%

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

Cancès

Zurek

2022

Numer. Math.

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“…We refer the reader to [27,5,31] for more details. We would like to highlight in particular the very recent work [12] where the authors propose a space-time Galerkin method which preserves the entropy structure of cross-diffusion systems, including the Stefan-Maxwell system under consideration.…”

Section: It Then Holds That For Allmentioning

confidence: 99%

Finite volumes for the Stefan-Maxwell cross-diffusion system

Cancès¹,

Ehrlacher²,

Monasse³

2020

Preprint

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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.

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An entropy structure preserving space-time formulation for cross-diffusion systems: Analysis and Galerkin discretization

Cited by 2 publications

References 43 publications

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

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

Finite volumes for the Stefan-Maxwell cross-diffusion system

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