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

Chun Liu, Cheng Wang, Steven M. Wise, Xingye Yue, Shenggao Zhou

arXiv:2009.08076v1math.NAcs.NA

TL;DR

The paper addresses the challenge of combining positivity and unique solvability, energy stability, and optimal-rate convergence for a PNP numerical scheme. It develops a variational finite-difference method and analyzes its properties. The analysis establishes positivity and solvability and provides an optimal-rate convergence result under stated regularity and mesh-refinement conditions.

  • Problem

    Existing PNP numerical analyses had not combined unique solvability and positivity, variational energy stability, and optimal-rate convergence in one scheme.

    Source: §1 · p. 3

  • Method

    The paper constructs a finite-difference scheme for the non-constant-mobility H−1 gradient-flow formulation, explicitly updating mobility and implicitly treating logarithmic and electric-potential diffusion terms.

    Source: §1 · p. 3, §7 · p. 27

  • Results

    The analysis establishes a unique positive numerical solution and optimal-rate convergence under sufficient regularity and sufficiently small time and space steps satisfying C1h ≤ ∆t ≤ C2h.

    Source: §3 · p. 7, §5 · p. 11

  • Takeaways & Limitations

    The scheme combines positivity and unique solvability, energy stability, and optimal-rate convergence for the PNP system.

    Source: §1 · p. 4, §7 · p. 27

    The analysis assumes periodic boundary conditions and zero background fixed charge, although the authors state that extensions are possible.

    Source: §1 · p. 2

Abstract

from arXiv · show

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 positivity-preserving 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 $\ell^\infty$ 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.

1 Introduction

The paper addresses the challenge of designing PNP schemes that combine solvability and positivity, energy stability, and optimal-rate convergence. It proposes a variational finite-difference scheme and outlines a convergence analysis for its non-constant-mobility structure.

1 Introduction

  • For simplicity, the analysis assumes periodic boundary conditions and zero background fixed charge; the authors state that extensions are possible.They describe extension to more physical boundary conditions and a nonzero source term as possible, with the latter straightforward.Source: §1 · p. 2
  • The PNP system is framed through EnVarA as an H−1-like gradient flow with non-constant mobility and chemical potentials.The passage introduces the system's gradient-flow formulation and chemical potentials.Source: §1 · pp. 2–3
  • The scheme explicitly updates mobility and treats logarithmic and electric-potential diffusion terms implicitly, using their convexity to establish solvability and energy stability.Positivity follows because the singular logarithmic energy prevents concentrations from reaching zero.Source: §1 · p. 3
  • The convergence analysis uses a higher-order asymptotic expansion, rough error bounds for n and p, and a refined error estimate to handle nonlinear parabolic coefficients.The expansion reaches third-order temporal and fourth-order spatial accuracy; the rough estimate provides ℓ∞ bounds.Source: §1 · pp. 3–4
  • The proposed scheme combines unique solvability and positivity preservation, energy stability in the variational framework, and optimal-rate convergence analysis.The authors describe this as the first such combination for a PNP numerical scheme.Source: §1 · pp. 3–4

2 The fully discrete numerical scheme

The fully discrete scheme is built from a dimensionless PNP gradient-flow formulation and a periodic finite-difference discretization. Its semi-implicit time update treats mobility explicitly and the chemical-potential terms implicitly.

2.1 Nondimensionalization

  • Nondimensionalization rescales concentrations and space and time, then writes the PNP dynamics with a dimensionless diffusion ratio.The dimensionless system is presented as dissipating energy and as a conserved gradient flow with dimensionless chemical potentials.Source: §2.1 · p. 4

2.2 The finite difference spatial discretization

  • The spatial discretization defines periodic cell- and face-centered grids, discrete difference and average operators, and associated inner products and norms.A uniform mesh is used for simplicity, and summation-by-parts formulas support the discrete analysis.Source: §2.2 · pp. 5–6

2.3 The numerical scheme

  • The semi-implicit scheme advances n and p using explicitly treated mobility and implicitly treated chemical-potential terms.The scheme is stated for the next time level from the current concentrations.Source: §2.3 · p. 6

3 Positivity-preserving and unique solvability analyses

The analysis establishes that each time step has a unique positive solution by representing the numerical scheme as minimization of a strictly convex discrete energy. The logarithmic singularity excludes zero concentrations, while convexity supports uniqueness.

3 Positivity-preserving and unique solvability analyses

  • For positive constant initial concentrations, the scheme has a unique solution at the next step with both concentrations strictly positive.The theorem assumes the preceding concentrations are equal and positive, with their difference in the mean-zero space.Source: §3 · p. 7
  • The discrete numerical solution is equivalent to minimizing an energy functional containing logarithmic concentration terms.The proof uses a domain bounded away from zero and an upper bound, then shows a minimizer cannot lie on the boundary when the lower cutoff is sufficiently small.Source: §3 · pp. 7–9
  • A minimizer in the interior yields a positive numerical solution equivalent to the scheme, and strict convexity gives uniqueness.The proof first establishes existence of a minimizer and then uses the interior minimum's stationarity conditions.Source: §3 · pp. 8–9

4 Energy stability analysis

The scheme’s discrete energy is nonincreasing at every time step, with unconditional energy stability established using the convexity of its energy terms.

4 Energy stability analysis

  • The discrete energy satisfies E_h(n^m, p^m) ≤ E_h(n^0, p^0) ≤ C_0 for all m, with C_0 independent of h.The initial energy bound follows from a consistency argument.Source: §4 · pp. 9–10
  • Convexity of the logarithmic and electric-potential energy terms supports the unconditional energy-stability proof.The proof takes discrete inner products of the scheme equations with the chemical potentials, then uses convexity to obtain the stability estimate.Source: §4 · pp. 9–10

5 Optimal rate convergence analysis in ℓ∞(0, T; ℓ2) ∩ℓ2(0, T; H1

The section proves optimal-rate convergence under regularity and mesh-refinement assumptions using higher-order consistency and error estimates. It establishes third-order temporal and fourth-order spatial convergence for the concentrations, recovers the a-priori assumption by induction for sufficiently small time step and mesh size, and gives a lower-order estimate for the electric potential.

5 Optimal rate convergence analysis in ℓ∞(0, T; ℓ2) ∩ℓ2(0, T; H1

  • Theorem 5.1 establishes convergence in the stated norms for sufficiently small steps, assuming C6 initial data, regularity class R, and C1h ≤ ∆t ≤ C2h.The estimate holds for tm = m∆t ≤ T, with C independent of ∆t and h; the supplied passage does not show the estimate’s explicit rate.Source: §5 · p. 11

5.1 Higher order consistency analysis of (2.10) – (2.13): asymptotic expansion of the numerical solution

  • A corrected asymptotic profile raises consistency to O(∆t^3+h^4), enabling an a-priori ℓ∞ bound needed to recover separation for the numerical solution.The ordinary projection solution has only first-order temporal and second-order spatial consistency, which is insufficient for that bound.Source: §5.1 · pp. 11, 15

5.2 A rough error estimate

  • The rough error estimate, under linear refinement, is used with an inverse inequality to establish uniform bounds and a separation property for the numerical solution.The estimate itself is lower order than the assumed prior error bound, so it supports the refined analysis rather than global induction.Source: §5.2 · p. 19

5.3 A refined error estimate

  • The refined error estimate begins with a preliminary lemma that assumes ℓ∞ bounds for the numerical solution at both the previous and next time steps.The supplied passage does not include the lemma’s resulting estimate.Source: §5.3 · p. 19
  • The refined concentration error is bounded by C(∆t^3 + h^4), yielding the convergence estimate for (n, p).The a-priori assumption is recovered at the next time step when ∆t and h are sufficiently small, enabling induction.Source: §5.3 · p. 24
  • The electric-potential error is bounded by a constant times ∆t + h^2.The estimate uses discrete elliptic regularity and the potential truncation error.Source: §5.3 · p. 24

6 Numerical results

The numerical tests report roughly second-order accuracy under mesh refinement and demonstrate discrete mass conservation, energy dissipation, and positive concentrations. They also show electrostatic screening over time and describe a relaxed iteration whose convergence analysis is deferred.

6 Numerical results

  • The nonlinear iteration uses a relaxation parameter, but its convergence to the proposed scheme is only expected under a sufficient time-step condition, with detailed analysis left for future work.Each iteration solves two linear systems for n and p associated with M-matrices.Source: §6 · p. 25

6.1 Accuracy tests

  • As the mesh is refined with ∆t = h^2, the ℓ∞ errors for p, n, and ψ decrease with convergence order about two.The accuracy tests use a known exact solution and source terms.Source: §6.1 · pp. 25–26, Table 1 · p. 26

6.2 Properties tests

  • Electrostatic interactions produce concentration peaks and valleys, while mobile ions of opposite charge quickly screen the initially induced potential; by T = 5, the system nearly reaches equilibrium.Snapshots are shown at T = 0.05, 0.1, and 5.Source: §6.2 · pp. 26–27, Figure 1 · p. 26
  • The discrete energy decreases monotonically, the total mass of cations is conserved, and the minimum concentration remains positive throughout the evolution.The tests therefore confirm mass conservation, energy dissipation, and positivity at the discrete level.Source: §6.2 · p. 27

7 Concluding remarks

The paper proposes a finite difference scheme for the PNP system and establishes positivity, energy stability, and optimal-rate convergence. The analysis relies on nonstandard estimates and higher-order asymptotic expansion to handle non-constant mobility.

7 Concluding remarks

  • The scheme explicitly treats mobility and implicitly treats logarithmic and electric-potential diffusion terms, supporting unique solvability and energy stability.The implicit treatment uses the convexity of the logarithmic and electric-potential energy terms.Source: §7 · p. 27
  • The logarithmic energy singularity near zero prevents either concentration from reaching zero, establishing positivity and keeping the scheme well-defined.The paper also establishes energy stability from the convex energy structure and positivity.Source: §7 · p. 27
  • Optimal-rate convergence is obtained using higher-order asymptotic expansion, rough error bounds for n and p, and subsequent refined error estimates.The expansion reaches third-order temporal and fourth-order spatial accuracy to support the rough bounds.Source: §7 · p. 27
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