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Convergence and long-time behavior of finite volumes for a generalized Poisson-Nernst-Planck system with cross-diffusion and size exclusion

Clément Cancès, Maxime Herda, Annamaria Massimini

Abstract

We present a finite volume scheme for modeling the diffusion of charged particles, specifically ions, in constrained geometries using a degenerate Poisson-Nernst-Planck system with size exclusion yielding cross-diffusion. Our method utilizes a two-point flux approximation and is part of the exponentially fitted scheme framework. The scheme is shown to be thermodynamically consistent, as it ensures the decay of some discrete version of the free energy. Classical numerical analysis results -- existence of discrete solution, convergence of the scheme as the grid size and the time step go to $0$ -- follow. We also investigate the long-time behavior of the scheme, both from a theoretical and numerical point of view. Numerical simulations confirm our findings, but also point out some possibly very slow convergence towards equilibrium of the system under consideration.

Convergence and long-time behavior of finite volumes for a generalized Poisson-Nernst-Planck system with cross-diffusion and size exclusion

Abstract

We present a finite volume scheme for modeling the diffusion of charged particles, specifically ions, in constrained geometries using a degenerate Poisson-Nernst-Planck system with size exclusion yielding cross-diffusion. Our method utilizes a two-point flux approximation and is part of the exponentially fitted scheme framework. The scheme is shown to be thermodynamically consistent, as it ensures the decay of some discrete version of the free energy. Classical numerical analysis results -- existence of discrete solution, convergence of the scheme as the grid size and the time step go to -- follow. We also investigate the long-time behavior of the scheme, both from a theoretical and numerical point of view. Numerical simulations confirm our findings, but also point out some possibly very slow convergence towards equilibrium of the system under consideration.

Paper Structure

This paper contains 21 sections, 16 theorems, 165 equations, 8 figures.

Table of Contents

  1. Presentation of the problem
  2. The continuous generalized Poisson--Nernst--Planck model
  3. Entropy structure of the model
  4. Weak solution
  5. Equilibrium and modified Poisson--Boltzmann equation
  6. Goal and positioning of the paper
  7. The finite volume scheme and main results
  8. Space and time discretizations
  9. The finite volume schemes
  10. Main results and organization of the paper
  11. Uniform a priori bounds and existence of a discrete solution
  12. Uniform a priori bounds and existence of a discrete solution
  13. Energy dissipation at the discrete level
  14. Further uniform estimates on the discrete solution
  15. Convergence of the schemes
  16. ...and 6 more sections

Key Result

Theorem 2.1

Given an admissible mesh $(\mathcal{T}, \mathcal{E}, (x_K)_{K\in\mathcal{T}})$ of $\Omega$ and a sequence of time steps $\left(\tau^n\right)_{n\geq 0}$ as in Section sec:space_time_discr, then there exists (at least) one solution to the scheme eq:scheme.Poisson--eq:uiK0 which satisfies Moreover, the discrete free energy $\mathcal{H}_\mathcal{T}^n$ defined later on in eq:free_energ_discr is decayi

Figures (8)

  • Figure 1: Examples of diamond cells $\omega_\sigma, \omega_{\sigma'}$ for inner and external faces $\sigma$ and $\sigma'$.
  • Figure 2: Left: Concentration profiles $u_1(T,x), u_2(T,x)$ and $u_0(T,x)$ at time $T = 1$ depicted with solid lines. The corresponding long-time limit $(u_1^\infty, u_2^\infty, u_3^\infty)$ is depicted with dashed lines. Right: Electric potential profile $\phi(T,x)$ at time $T = 1$ (solid line) and in the long-time limit (dashed line).
  • Figure 3: Convergence of the schemes under space grid refinement.
  • Figure 4: Geometry of the two-dimensional domain
  • Figure 5: Stationary concentration profiles corresponding to the initialization \ref{['eq:num.u0.3']} for $\lambda^2 = 0.01$
  • ...and 3 more figures

Theorems & Definitions (31)

  • Definition 1.1: Weak solution
  • Theorem 2.1
  • Theorem 2.2
  • Theorem 2.3
  • Lemma 3.1: Mass conservation
  • proof
  • Proposition 3.2: Positivity of volume fractions
  • proof
  • Proposition 3.3: Uniform bound for the electric potential
  • Proposition 3.4: Existence of solutions
  • ...and 21 more