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

Huo, Xiaokai; Liu, Hailiang; Tzavaras, Athanasios E.; Wang, Shuaikun

doi:10.1137/20m1338666

Cited by 7 publications

(1 citation statement)

References 26 publications

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“…In order to extend the above JKO-type scheme to the present setting, we face two new difficulties: (i) D i (x) is no longer a constant, the kinetic energy corresponding to the squared distance cost needs to be modified; (ii) ǫ(x) is a general non-negative function, φ cannot be expressed explicitly in terms of ρ. As for (i), we follow [16] and consider a modified functional…”

Section: Model Background and Semi-discretizationmentioning

confidence: 99%

A dynamic mass transport method for Poisson-Nernst-Planck equations

Liu,

Maimaitiyiming

2022

Preprint

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A dynamic mass-transport method is proposed for approximately solving the Poisson-Nernst-Planck (PNP) equations. The semi-discrete scheme based on the JKO type variational formulation naturally enforces solution positivity and the energy law as for the continuous PNP system. The fully discrete scheme is further formulated as a constrained minimization problem, shown to be solvable, and satisfy all three solution properties (mass conservation, positivity and energy dissipation) independent of time step size or the spatial mesh size. Numerical experiments are conducted to validate convergence of the computed solutions and verify the structure preserving property of the proposed scheme.

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Section: Model Background and Semi-discretizationmentioning

confidence: 99%

A dynamic mass transport method for Poisson-Nernst-Planck equations

Liu,

Maimaitiyiming

2022

Preprint

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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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A dynamic mass transport method for Poisson-Nernst-Planck equations

Liu

Maimaitiyiming

2023

Journal of Computational Physics

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

Cited by 7 publications

References 26 publications

A dynamic mass transport method for Poisson-Nernst-Planck equations

A dynamic mass transport method for Poisson-Nernst-Planck equations

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

A dynamic mass transport method for Poisson-Nernst-Planck equations

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