Entropy and convergence analysis for two finite volume schemes for a Nernst–Planck–Poisson system with ion volume constraints

Gaudeul, Benoît; Fuhrmann, Juergen

doi:10.1007/s00211-022-01279-y

Cited by 7 publications

(4 citation statements)

References 41 publications

(62 reference statements)

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“…The results established in this last work have been recently generalized in [64] by employing a two-point flux approximation finite volume scheme with mobilities given by logarithmic mean ensuring the preservation of the entropy structure of the models exhibited in [39]. We also refer to [37] where an entropy preserving finite volume scheme is analyzed for a spinorial matrix drift-diffusion model for semiconductors, and to [49] where two energy stable two-point flux approximation finite volume schemes are studied for a generalized Poisson-Nernst-Planck system accounting for the excess chemical potential of the solvent in multicomponent ionic liquids.…”

Section: 3mentioning

confidence: 99%

“…in terms of discrete unknowns approximating (u, µ) rather than (u, π). A typical example of such approaches is the widely used upstream mobility scheme (see for instance [46,26]) or smoothened counterparts taking advantage of the self-diffusion (see for instance [19,49]). Such schemes would naturally be energy diminishing in accordance with the gradient flow structure highlighted in Section 2.1.…”

Section: Notation and Definitions We Present The Discretization Of Th...mentioning

confidence: 99%

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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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Section: 3mentioning

confidence: 99%

Section: Notation and Definitions We Present The Discretization Of Th...mentioning

confidence: 99%

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

Cancès

Zurek

2022

Numer. Math.

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“…This model was used into Li et al 17,40 to describe the initial stage of the scintillation, before the recombination takes place. The system ( 87) is well-studied, and there are many results, dealing with existence, asymptotic decay, equilibrium solutions and even explicit analytical solutions for k = 2 (e.g., among the many [86][87][88][89] and for more recent advances [90][91][92][93] ). Equation (87) 1 can be put in the equivalent gradient flow formulation (21) with K = K D .…”

Section: The Diffusion-drift Approximationmentioning

confidence: 99%

A brief overview of existence results and decay time estimates for a mathematical modeling of scintillating crystals

Davı́

2021

Math Methods in App Sciences

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Inorganic scintillating crystals can be modelled as continua with microstructure.For rigid and isothermal crystals, the evolution of charge carriers becomes in this way described by a reaction-diffusion-drift equation coupled with the Poisson equation of electrostatic. Here, we give a survey of the available existence and asymptotic decays results for the resulting boundary value problem, the latter being a direct estimate of the scintillation decay time. We also show how to recover various approximated models which encompass also the two most used phenomenological models for scintillators, namely, the kinetic and diffusive ones. Also for these cases, we show, whenever it is possible, which existence and asymptotic decays estimate results are known to date. KEYWORDS entropy methods, existence of solutions of PDE, exponential rate of convergence, reaction-diffusion-drift equations, scintillators MSC CLASSIFICATION 35K57; 35B40; 35B45 INTRODUCTIONA scintillator crystal is a material which converts ionizing radiations into photons in the frequency range of visible light, hence its name. It acts as a true "wavelength shifter" and in such a role is used as radiation sensor into high-energy physics, in medical imaging and in security applications. 1 The physics of scintillation, which is a complex multi-scale phenomenon (see, e.g., Vasil'ev and Getkin 2 ) can be described within a continuum approach at three scales: at a microscopic scale, the incoming energy E generates a population of charged energy carriers which moves in straight directions for few nanometer 3 and whose density N = N(E) can be found by the means of approximated solutions of the Bethe-Bloch equation. 4-7* These energy carriers wander and migrate within a greater region either generating other energy carriers or recombining with emission of photons h𝜈. In the process, some energy is lost, and a scintillator is a material in which

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“…The model and the scheme originate from [3]. The scheme relies on the the excess chemical potential flux scheme which appears to be used for the first time in [40] and was later numerically analyzed in [11,23] and compared in [1,28].…”

Section: Introductionmentioning

confidence: 99%

Numerical analysis of a finite volume scheme for charge transport in perovskite solar cells

Abdel¹,

Chainais-Hillairet²,

Farrell³

et al. 2022

Preprint

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In this paper, we consider a drift-diffusion charge transport model for perovskite solar cells, where electrons and holes may diffuse linearly (Boltzmann approximation) or nonlinearly (e.g. due to Fermi-Dirac statistics). To incorporate volume exclusion effects, we rely on the Fermi-Dirac integral of order −1 when modeling moving anionic vacancies within the perovskite layer which is sandwiched between electron and hole transport layers. After nondimensionalization, we first prove a continuous entropy-dissipation inequality for the model. Then, we formulate a corresponding two-point flux finite volume scheme on Voronoi meshes and show an analogous discrete entropy-dissipation inequality. This inequality helps us to show the existence of a discrete solution of the nonlinear discrete system with the help of a corollary of Brouwer's fixed point theorem and the minimization of a convex functional. Finally, we verify our theoretically proven properties numerically, simulate a realistic device setup and show exponential decay in time with respect to the L 2 error as well as a physically and analytically meaningful relative entropy.

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Entropy and convergence analysis for two finite volume schemes for a Nernst–Planck–Poisson system with ion volume constraints

Cited by 7 publications

References 41 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

A brief overview of existence results and decay time estimates for a mathematical modeling of scintillating crystals

Numerical analysis of a finite volume scheme for charge transport in perovskite solar cells

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