A Convergent Entropy Diminishing Finite Volume Scheme for a Cross-Diffusion System

Cancès, Clément; Gaudeul, Benoît

doi:10.1137/20m1316093

Cited by 16 publications

(18 citation statements)

References 45 publications

(57 reference statements)

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“…Compactness properties. Let (D m ) m∈N be a sequence of admissible meshes of Ω T satisfying the mesh regularity (8) uniformly in m ∈ N and let ∆t m < 1/C f . We claim that the estimates from Lemmas 7 and 8 imply the strong convergence of a subsequence of (u i,m ).…”

Section: Proof Of Theoremmentioning

confidence: 99%

“…We take rather stiff values of the Lotka-Volterra constants as in [ Since exact solutions to the SKT model are not explicitly known, we compute a reference solution on a uniform mesh composed of 5120 cells and with ∆t = (1/5120) 2 . We use this rather small value of ∆t because the Euler discretization in time exhibits a first-order convergence rate, while we expect, as observed for instance in [8], a second-order convergence rate in space for scheme (9)-( 12), due to the logarithmic mean used to approximate the mobility coefficients in the numerical fluxes. We compute approximate solutions on uniform meshes made of 40, 80, 160, 320, 640, and 1280 cells, respectively.…”

Section: Secnumermentioning

confidence: 99%

“…Convergence of the scheme). Let the assumptions of Theorem 1 hold, let (D m ) m∈N be a family of admissible meshes satisfying(8) uniformly in m ∈ N, and assume that ∆t m < 1/C f for m ∈ N. Let (u m ) m∈N be a family of finite-volume solutions to (9)-(12) constructed in Theorem 1. Then there exists a function u = (u 1 , .…”

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confidence: 99%

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A convergent structure-preserving finite-volume scheme for the Shigesada-Kawasaki-Teramoto population system

Zurek¹,

Jüngel²

2020

Preprint

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An implicit Euler finite-volume scheme for an n-species population crossdiffusion system of Shigesada-Kawasaki-Teramoto-type in a bounded domain with no-flux boundary conditions is proposed and analyzed. The scheme preserves the formal gradientflow or entropy structure and preserves the nonnegativity of the population densities. The key idea is to consider a suitable mean of the mobilities in such a way that a discrete chain rule is fulfilled and a discrete analog of the entropy inequality holds. The existence of finite-volume solutions, the convergence of the scheme, and the large-time asymptotics to the constant steady state are proven. Furthermore, numerical experiments in one and two space dimensiona for two and three species are presented. The results are valid for a more general class of cross-diffusion systems satisfying some structural conditions.

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Section: Proof Of Theoremmentioning

confidence: 99%

Section: Secnumermentioning

confidence: 99%

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A convergent structure-preserving finite-volume scheme for the Shigesada-Kawasaki-Teramoto population system

Zurek¹,

Jüngel²

2020

Preprint

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“…A fully implicit Euler--Galerkin scheme is developed in [17] for the Maxwell--Stefan system coupled with a Poisson equation, which is positivity-preserving, energy-stable, and convergent. Recently, in [5], an implicit finite volume scheme was proposed for a cross-diffusion system similar to the Maxwell--Stefan system. A nonlinear cutoff function was used to approximate the values at cell interfaces to ensure nonnegativity of solutions.…”

mentioning

confidence: 99%

“…A nonlinear cutoff function was used to approximate the values at cell interfaces to ensure nonnegativity of solutions. Both schemes in [17] and [5] incorporate the entropy structure to ensure the energy-stable property. The scheme proposed here is positivity-preserving and entropy-decreasing and provides a connection between the finite difference scheme and a variational minimization problem.…”

mentioning

confidence: 99%

An Energy Stable and Positivity-Preserving Scheme for the Maxwell--Stefan Diffusion System

Huo¹,

Liu²,

Tzavaras³

et al. 2021

SIAM J. Numer. Anal.

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We develop a new finite difference scheme for the Maxwell--Stefan diffusion system. The scheme is conservative, energy-stable, and positivity-preserving. These nice properties stem from a variational structure and are proved by reformulating the finite difference scheme into an equivalent optimization problem. The solution to the scheme emerges as the minimizer of the optimization problem, and as a consequence energy stability and positivity-preserving properties are obtained.

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

Cancès,

Cauvin-Vila,

Chainais-Hillairet

et al. 2023

Springer Proceedings in Mathematics &Amp; Statistics

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We propose and study a one-dimensional model which consists of two cross-diffusion systems coupled via a moving interface. The motivation stems from the modelling of complex diffusion processes in the context of the vapor deposition of thin films. In our model, crossdiffusion of the various chemical species can be respectively modelled by a size-exclusion system for the solid phase and the Stefan-Maxwell system for the gaseous phase. The coupling between the two phases is modelled by linear phase transition laws of Butler-Volmer type, resulting in an interface evolution. The continuous properties of the model are investigated, in particular its entropy variational structure and stationary states. We introduce a two-point flux approximation finite volume scheme. The moving interface is addressed with a moving-mesh approach, where the mesh is locally deformed around the interface. The resulting discrete nonlinear system is shown to admit a solution that preserves the main properties of the continuous system, namely: mass conservation, nonnegativity, volume-filling constraints, decay of the free energy and asymptotics. In particular, the moving-mesh approach is compatible with the entropy structure of the continuous model. Numerical results illustrate these properties and the dynamics of the model.

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A Convergent Entropy Diminishing Finite Volume Scheme for a Cross-Diffusion System

Cited by 16 publications

References 45 publications

A convergent structure-preserving finite-volume scheme for the Shigesada-Kawasaki-Teramoto population system

A convergent structure-preserving finite-volume scheme for the Shigesada-Kawasaki-Teramoto population system

An Energy Stable and Positivity-Preserving Scheme for the Maxwell--Stefan Diffusion System

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

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