A Positivity Preserving and Free Energy Dissipative Difference Scheme for the Poisson–Nernst–Planck System

He, Dongdong; Pan, Kejia; Yue, Xiaoqiang

doi:10.1007/s10915-019-01025-x

Cited by 30 publications

(11 citation statements)

References 38 publications

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“…More recent attempts have focused on semi-implicit schemes based on a formulation of the nonlogarithmic Landau type. As a result, all schemes obtained in [5,15,16,22,23] have been shown to feature unconditional positivity ( see further discussion in section 1.2).…”

Section: Introductionmentioning

confidence: 91%

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Efficient, positive, and energy stable schemes for multi-D Poisson-Nernst-Planck systems

Liu

Maimaitiyiming

2020

Preprint

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In this paper, we design, analyze, and numerically validate positive and energydissipating schemes for solving the time-dependent multi-dimensional system of Poisson-Nernst-Planck (PNP) equations, which has found much use in the modeling of biological membrane channels and semiconductor devices. The semi-implicit time discretization based on a reformulation of the system gives a well-posed elliptic system, which is shown to preserve solution positivity for arbitrary time steps. The first order (in time) fully-discrete scheme is shown to preserve solution positivity and mass conservation unconditionally, and energy dissipation with only a mild O(1) time step restriction. The scheme is also shown to preserve the steady-states. For the fully second order (in both time and space) scheme with large time steps, solution positivity is restored by a local scaling limiter, which is shown to maintain the spatial accuracy. These schemes are easy to implement. Several three-dimensional numerical examples verify our theoretical findings and demonstrate the accuracy, efficiency, and robustness of the proposed schemes, as well as the fast approach to steady states.

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Section: Introductionmentioning

confidence: 91%

“…with a consistent choice for ψ * i and integer k ≥ 1. Different options are introduced in [5,15,16] for obtaining their respective positive schemes.…”

Section: Our Contributionsmentioning

confidence: 99%

Efficient, positive, and energy stable schemes for multi-D Poisson-Nernst-Planck systems

Liu

Maimaitiyiming

2020

Preprint

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“…Recent efforts have been on the design of efficient and stable methods with structurepreserving analysis. On regular domains, results using finite difference/volume for spatial discretization are quite rich, including the works [3,6,7,10,11,13,22,23,24,37], as we discussed above. On irregular domains, Mirzadeh et al [34] presented a conservative hybrid method with adaptive strategies.…”

Section: Further Related Workmentioning

confidence: 99%

“…This is evidenced by recent results in [6,10,24] with second order finite difference schemes. Based on some formulations of the nonlogarithmic Landau type (see (1.2) below), the semi-implicit schemes in [3,11,13,22,23] have been shown to feature unconditional positivity, while the energy dissipation is handled differently. For instance, the unconditional positive schemes in [23] are linear, and shown to feature energy dissipation with only an O(1) time step restriction.…”

Section: Introductionmentioning

confidence: 99%

Positivity-preserving third order DG schemes for Poisson--Nernst--Planck equations

Liu,

Wang,

Yin

et al. 2021

Preprint

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In this paper, we design and analyze third order positivity-preserving discontinuous Galerkin (DG) schemes for solving the time-dependent system of Poisson-Nernst-Planck (PNP) equations, which has found much use in diverse applications. Our DG method with Euler forward time discretization is shown to preserve the positivity of cell averages at all time steps. The positivity of numerical solutions is then restored by a scaling limiter in reference to positive weighted cell averages. The method is also shown to preserve steady states. Numerical examples are presented to demonstrate the third order accuracy and illustrate the positivity-preserving property in both one and two dimensions.

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“…A three-level linearized difference scheme [26][27][28][29] for boundary value problem (1)- (3) with homogeneous Dirichlet boundary conditions (12) in the finite domain Ω = [a, b] is as follows:…”

Section: Numerical Schemementioning

confidence: 99%

A fourth-order linearized difference scheme for the coupled space fractional Ginzburg–Landau equation

Yuan

Zeng

2019

Adv Differ Equ

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In this paper, the coupled space fractional Ginzburg-Landau equations are investigated numerically. A linearized semi-implicit difference scheme is proposed. The scheme is unconditionally stable, fourth-order accurate in space, and second-order accurate in time. The optimal pointwise error estimates, unique solvability, and unconditional stability are obtained. Moreover, Richardson extrapolation is exploited to improve the temporal accuracy to fourth order. Finally, numerical results are presented to confirm the theoretical results.

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A Positivity Preserving and Free Energy Dissipative Difference Scheme for the Poisson–Nernst–Planck System

Cited by 30 publications

References 38 publications

Efficient, positive, and energy stable schemes for multi-D Poisson-Nernst-Planck systems

Efficient, positive, and energy stable schemes for multi-D Poisson-Nernst-Planck systems

Positivity-preserving third order DG schemes for Poisson--Nernst--Planck equations

A fourth-order linearized difference scheme for the coupled space fractional Ginzburg–Landau equation

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