Optimal Rate Convergence Analysis of a Second Order Numerical Scheme for the Poisson-Nernst-Planck System

Ding, Jie

doi:10.4208/nmtma.oa-2018-0058

Cited by 17 publications

(9 citation statements)

References 25 publications

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“…Notice that the definition of the boundary data ψ D and σ have been extended to the whole computational domain. In numerical implementation, the ghost points outside Ω are eliminated by coupling the discretization scheme ( 8) and boundary discretization ( 9) and (10). The coupled difference equations can be written in a matrix form…”

Section: Finite Difference Methods In Space 1) Spatial Discretization...mentioning

confidence: 99%

Structure-Preserving and Efficient Numerical Methods for Ion Transport

Ding,

Wang,

Zhou

2020

Preprint

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Ion transport, often described by the Poisson-Nernst-Planck (PNP) equations, is ubiquitous in electrochemical devices and many biological processes of significance. In this work, we develop conservative, positivity-preserving, energy dissipating, and implicit finite difference schemes for solving the multi-dimensional PNP equations with multiple ionic species. A central-differencing discretization based on harmonic-mean approximations is employed for the Nernst-Planck (NP) equations. The backward Euler discretization in time is employed to derive a fully implicit nonlinear system, which is efficiently solved by a newly proposed Newton's method. The improved computational efficiency of the Newton's method originates from the usage of the electrostatic potential as the iteration variable, rather than the unknowns of the nonlinear system that involves both the potential and concentration of multiple ionic species. Numerical analysis proves that the numerical schemes respect three desired analytical properties (conservation, positivity preserving, and energy dissipation) fully discretely. Based on advantages brought by the harmonic-mean approximations, we are able to establish estimate on the upper bound of condition numbers of coefficient matrices in linear systems that are solved iteratively. The solvability and stability of the linearized problem in the Newton's method are rigorously established as well. Numerical tests are performed to confirm the anticipated numerical accuracy, computational efficiency, and structure-preserving properties of the developed schemes. Adaptive time stepping is implemented for further efficiency improvement. Finally, the proposed numerical approaches are applied to characterize ion transport subject to a sinusoidal applied potential.

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Section: Finite Difference Methods In Space 1) Spatial Discretization...mentioning

confidence: 99%

Structure-Preserving and Efficient Numerical Methods for Ion Transport

Ding,

Wang,

Zhou

2020

Preprint

Self Cite

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“…), (5.82) in which the a-priori estimates (5.53), (5.55) have been used in the second step, and the following inverse inequality has been applied in the last step: 4 ).…”

Section: A Rough Error Estimatementioning

confidence: 99%

“…Such a stability analysis has appeared in a few existing numerical works [11,23,27], while the unique solvability and positivity-preserving analysis have been missing. Furthermore, there have been a few existing works for the convergence analysis [4,31,35], while these convergence estimates have been based on the perfect Laplacian operator structure for n and p, instead of the H −1 gradient flow structure, so that the energy estimate is not available. Many other numerical schemes have been reported [13,16,27,28,33,36,40].…”

Section: Introductionmentioning

confidence: 99%

“…We provide an optimal rate convergence analysis for the proposed numerical scheme. The variational structure and the non-constant mobility make this analysis highly challenging, especially when compared with existing convergence estimates in [4,31,35], wherein a perfect Laplacian operator is kept in tact. To overcome such a well-known difficulty, several highly non-standard estimates have to be introduced, due to the nonlinear parabolic coefficients.…”

Section: Introductionmentioning

confidence: 99%

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A positivity-preserving, energy stable and convergent numerical scheme for the Poisson-Nernst-Planck system

Liu

Wang

Wise³

et al. 2020

Preprint

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In this paper we propose and analyze a finite difference numerical scheme for the Poisson-Nernst-Planck equation (PNP) system. To understand the energy structure of the PNP model, we make use of the Energetic Variational Approach (EnVarA), so that the PNP system could be reformulated as a non-constant mobility H −1 gradient flow, with singular logarithmic energy potentials involved. To ensure the unique solvability and energy stability, the mobility function is explicitly treated, while both the logarithmic and the electric potential diffusion terms are treated implicitly, due to the convex nature of these two energy functional parts. The positivitypreserving property for both concentrations, n and p, is established at a theoretical level. This is based on the subtle fact that the singular nature of the logarithmic term around the value of 0 prevents the numerical solution reaching the singular value, so that the numerical scheme is always well-defined. In addition, an optimal rate convergence analysis is provided in this work, in which many highly non-standard estimates have to be involved, due to the nonlinear parabolic coefficients. The higher order asymptotic expansion (up to third order temporal accuracy and fourth order spatial accuracy), the rough error estimate (to establish the ∞ bound for n and p), and the refined error estimate have to be carried out to accomplish such a convergence result. In our knowledge, this work will be the first to combine the following three theoretical properties for a numerical scheme for the PNP system: (i) unique solvability and positivity, (ii) energy stability, and (iii) optimal rate convergence. A few numerical results are also presented in this article, which demonstrates the robustness of the proposed numerical scheme.

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“…We focus on the development of numerical methods for the PNPCH equations. Many efforts have been devoted to the development of numerical methods for the PNP-type equations, ranging from finite difference schemes to discontinuous Galerkin (DG) methods [9,10,16,18,37,41,42,50]. In order to obtain physically faithful numerical solutions, it is highly desirable and crucial to preserve physical properties of the analytical solutions, such as mass conservation, free-energy dissipation, and positivity, at the discrete level.…”

Section: Introductionmentioning

confidence: 99%

A Positive and Energy Stable Numerical Scheme for the Poisson-Nernst-Planck-Cahn-Hilliard Equations with Steric Interactions

Qian,

Wang,

Zhou

2020

Preprint

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In this work, we consider numerical methods for the Poisson-Nernst-Planck-Cahn-Hilliard (PNPCH) equations with steric interactions, which correspond to an H −1 gradient flow of a free-energy functional that consists of electrostatic free energies, steric interaction energies of short range, entropic contribution of ions, and concentration gradient energies. We propose a novel energy stable numerical scheme that respects mass conservation and positivity at the discrete level. Existence and uniqueness of the solution to the proposed nonlinear scheme are established by showing that the solution is a unique minimizer of a convex functional over a closed, convex domain. The positivity of numerical solutions is further theoretically justified by the singularity of the entropy terms, which prevents the minimizer from approaching zero concentrations. A further numerical analysis proves discrete free-energy dissipation. Extensive numerical tests are performed to validate that the numerical scheme is first-order accurate in time and second-order accurate in space, and is capable of preserving the desired properties, such as mass conservation, positivity, and free energy dissipation, at the discrete level. Moreover, the PNPCH equations and the proposed scheme are applied to study charge dynamics and self-assembled nanopatterns in highly concentrated electrolytes that are widely used in electrochemical energy devices. Numerical results demonstrate that the PNPCH equations and our numerical scheme are able to capture nanostructures, such as lamellar patterns and labyrinthine patterns in electric double layers and the bulk, and multiple time relaxation with multiple time scales. The multiple time relaxation dynamics with metastability take long time to reach an equilibrium, highlighting the need for robust, energy stable numerical schemes that allow large time stepping. In addition, we numerically characterize the interplay between cross steric interactions of short range and the concentration gradient regularization, and their impact on the development of nanostructures in the equilibrium state.

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Optimal Rate Convergence Analysis of a Second Order Numerical Scheme for the Poisson-Nernst-Planck System

Cited by 17 publications

References 25 publications

Structure-Preserving and Efficient Numerical Methods for Ion Transport

Structure-Preserving and Efficient Numerical Methods for Ion Transport

A positivity-preserving, energy stable and convergent numerical scheme for the Poisson-Nernst-Planck system

A Positive and Energy Stable Numerical Scheme for the Poisson-Nernst-Planck-Cahn-Hilliard Equations with Steric Interactions

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