pith. sign in

arxiv: 2604.08228 · v1 · submitted 2026-04-09 · 🧮 math.NA · cs.NA

Five-Structures Preserving Algorithm for charge dynamics model

Haoran Sun , Wancheng Wu , Kun Wang This is my paper

Pith reviewed 2026-05-10 17:35 UTC · model grok-4.3

classification 🧮 math.NA cs.NA
keywords structure-preserving schemesMaxwell-Ampere Nernst-Planck equationspositivity preservationGauss lawFaraday lawfinite difference methodsion transportcharge conservation
0
0 comments X

The pith

Numerical schemes for the Maxwell-Ampere Nernst-Planck equations exactly preserve mass conservation, positivity, energy dissipation, Gauss's law, and Faraday's law.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper develops numerical algorithms for simulating charged particle movement governed by the nonlinear Maxwell-Ampere Nernst-Planck system. It applies the Slotboom transformation to rewrite the equations for positivity, then uses backward Euler or BDF2 time stepping with finite differences, followed by a displacement correction to enforce Gauss's law and potential reconstruction to enforce Faraday's law. These steps produce fully discrete schemes that satisfy five structures exactly: mass conservation, concentration positivity, energy dissipation, Gauss's law, and Faraday's law, along with error estimates. The exact preservations matter because they prevent artificial drift or unphysical results in long-time ion transport simulations, as confirmed by tests on analytical solutions and fixed-charge cases.

Core claim

The central claim is that a family of schemes for the nonlinear Maxwell-Ampere Nernst-Planck equations exactly satisfies mass conservation, concentration positivity, energy dissipation, Gauss's law, and Faraday's law at the fully discrete level. The first-order version rewrites the Nernst-Planck part via Slotboom transformation, discretizes with backward Euler and centered differences, then applies displacement correction and potential reconstruction. The second-order version replaces the time step with BDF2 while retaining the same correction strategies. Both versions come with established error estimates, and experiments verify convergence orders, positivity, and exact law preservation in

What carries the argument

Slotboom transformation to enable positivity, together with displacement correction for Gauss's law and potential reconstruction for Faraday's law in the discretized system.

If this is right

  • The schemes keep all ion concentrations nonnegative at every time step.
  • Total mass of each species remains exactly constant throughout the evolution.
  • The discrete energy decreases monotonically in accordance with the continuous dissipation law.
  • Gauss's law holds exactly after each correction step without extra projections.
  • Faraday's law is satisfied exactly, allowing faithful reproduction of magnetic field effects in long simulations.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The exact preservations could support stable long-time runs of biological or electrochemical systems without gradual violation of physical constraints.
  • The same correction ideas might extend to other electro-diffusion models where multiple conservation laws must hold simultaneously.
  • Numerical tests already show the schemes reproduce ion accumulation and electric screening, suggesting they can capture realistic transport behavior accurately.

Load-bearing premise

The displacement correction and potential reconstruction steps maintain the five structures without introducing unaccounted truncation errors in the nonlinear coupled system.

What would settle it

A run of the first-order scheme on the analytical solution test case where the discrete mass of any ion species changes by more than machine precision or a concentration turns negative would falsify the exact preservation claim.

Figures

Figures reproduced from arXiv: 2604.08228 by Haoran Sun, Kun Wang, Wancheng Wu.

Figure 1
Figure 1. Figure 1: Correction schematic: progressive correction from [PITH_FULL_IMAGE:figures/full_fig_p007_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: The concentration 1, ℎ at different times in Example 2 [PITH_FULL_IMAGE:figures/full_fig_p021_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: The concentration 2, ℎ at different times in Example 2 As shown in Figures 2 and 3, the concentrations of the two ions evolve from a uniform initial state. Driven by electrostatic attraction, ions accumulate at locations with fixed charges opposite to their own sign, forming distinct concentration patterns. Over time, positive and negative ions gradually accumulate around the fixed charge regions, the conc… view at source ↗
Figure 4
Figure 4. Figure 4: The electric displacements D ℎ at different times in Example 2 [PITH_FULL_IMAGE:figures/full_fig_p023_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: The potential [PITH_FULL_IMAGE:figures/full_fig_p024_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: Gauss’s law in Example 2 From [PITH_FULL_IMAGE:figures/full_fig_p025_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: Faraday’s law in Example 2 From [PITH_FULL_IMAGE:figures/full_fig_p025_7.png] view at source ↗
Figure 8
Figure 8. Figure 8: Positivity of ion concentrations in Example 2 [PITH_FULL_IMAGE:figures/full_fig_p026_8.png] view at source ↗
Figure 9
Figure 9. Figure 9: Mass conservation in Example 2 As shown in [PITH_FULL_IMAGE:figures/full_fig_p026_9.png] view at source ↗
Figure 10
Figure 10. Figure 10: Energy dissipation in Example 2 As shown in [PITH_FULL_IMAGE:figures/full_fig_p027_10.png] view at source ↗
Figure 11
Figure 11. Figure 11: The concentration 1, ℎ at different times in Example 3 [PITH_FULL_IMAGE:figures/full_fig_p028_11.png] view at source ↗
Figure 12
Figure 12. Figure 12: The concentration 2, ℎ at different times in Example 3 As shown in Figures 11 and 12, the two ion concentrations 1 ℎ and 2 ℎ evolve from a uniform initial state, forming crescent-shaped patterns under the combined action of electrostatic attraction and chemical potential gradients. Positively charged 1 ℎ accumulates in the negative fixed charge region, while negatively charged 2 ℎ accumulates in the posit… view at source ↗
Figure 13
Figure 13. Figure 13: The electric displacements D ℎ at different times in Example 3 [PITH_FULL_IMAGE:figures/full_fig_p030_13.png] view at source ↗
Figure 14
Figure 14. Figure 14: The potential [PITH_FULL_IMAGE:figures/full_fig_p031_14.png] view at source ↗
Figure 15
Figure 15. Figure 15: Gauss’s law in Example 3 [PITH_FULL_IMAGE:figures/full_fig_p032_15.png] view at source ↗
Figure 16
Figure 16. Figure 16: Faraday’s law in Example 3 From [PITH_FULL_IMAGE:figures/full_fig_p032_16.png] view at source ↗
Figure 17
Figure 17. Figure 17: Positivity of ion concentrations in Example 3 [PITH_FULL_IMAGE:figures/full_fig_p033_17.png] view at source ↗
Figure 18
Figure 18. Figure 18: Mass conservation in Example 3 From [PITH_FULL_IMAGE:figures/full_fig_p033_18.png] view at source ↗
Figure 19
Figure 19. Figure 19: Energy dissipation in Example 3 [PITH_FULL_IMAGE:figures/full_fig_p034_19.png] view at source ↗
read the original abstract

This paper develops a family of fast, structure-preserving numerical algorithms for the nonlinear Maxwell-Ampere Nernst-Planck equations. For the first-order scheme, the Slotboom transformation rewrites the Nernst-Planck equation to enable positivity preservation. The backward Euler method and centered finite differences discretize the transformed system. Two correction strategies are introduced: one enforces Gauss's law via a displacement correction, and the other preserves Faraday's law through potential reconstruction. The fully discrete scheme exactly satisfies mass conservation, concentration positivity, energy dissipation, Gauss's law, and Faraday's law, with established error estimates. The second-order scheme adopts BDF2 time discretization while retaining the same structure-preserving strategies, exactly conserving mass, Gauss's law, and Faraday's law. Numerical experiments validate both schemes using analytical solutions, confirming convergence orders and positivity preservation. Simulations of ion transport with fixed charges demonstrate exact preservation of Gauss's and Faraday's laws over long-time evolution, reproducing electrostatic attraction, ion accumulation, and electric field screening. The results fully support the theoretical analysis and the schemes' stability and superior performance.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

3 major / 2 minor

Summary. The manuscript develops first- and second-order structure-preserving schemes for the nonlinear Maxwell-Ampere Nernst-Planck equations. The first-order scheme applies the Slotboom transformation, backward Euler discretization, and centered differences, followed by a displacement correction to enforce Gauss's law and potential reconstruction to enforce Faraday's law; it claims exact discrete preservation of mass conservation, concentration positivity, energy dissipation, Gauss's law, and Faraday's law together with error estimates. The second-order scheme uses BDF2 time stepping while retaining the same corrections but claims only mass, Gauss's law, and Faraday's law preservation. Numerical experiments with manufactured solutions and fixed-charge ion transport simulations are used to confirm convergence orders and long-time structure preservation.

Significance. If the exact simultaneous preservation of all five structures after the post-discretization corrections can be rigorously established in the nonlinear coupled system, the work would offer a useful advance for stable long-time simulations in electrodiffusion and plasma modeling. The explicit construction of corrections that target multiple conservation laws and the reported numerical confirmation of exact Gauss/Faraday preservation over extended times are concrete strengths.

major comments (3)
  1. [Abstract] Abstract: the central claim that the first-order scheme exactly satisfies energy dissipation and positivity after the displacement correction (to enforce Gauss's law) and potential reconstruction (to enforce Faraday's law) is load-bearing, yet the abstract provides no indication that the energy equality or positivity bound survives the field and concentration alterations introduced by these corrections in the fully nonlinear setting. The fact that the BDF2 variant drops both energy and positivity claims suggests the corrections are not neutral with respect to those structures.
  2. [Description of the correction strategies] Description of the correction strategies: because the base discretization is constructed to preserve positivity and energy dissipation while the corrections are applied afterward, an explicit verification (or counter-example) is required to show that the altered discrete electric field and concentrations still obey the exact discrete energy law and keep concentrations non-negative; without this step the five-structure claim cannot be accepted at face value.
  3. [Numerical experiments section] Numerical experiments section: the reported tests confirm exact Gauss/Faraday preservation and convergence orders, but do not appear to include quantitative monitoring of the discrete energy dissipation rate or positivity margin after the corrections under strong nonlinear coupling; such diagnostics would be necessary to support the theoretical claims.
minor comments (2)
  1. The abstract could briefly state the precise form of the Slotboom transformation and the correction operators to improve readability for readers unfamiliar with the specific discretization.
  2. A short table summarizing which structures are preserved by each scheme (first-order vs. BDF2) would clarify the differences at a glance.

Simulated Author's Rebuttal

3 responses · 0 unresolved

We thank the referee for the careful reading and constructive comments, which help clarify the presentation of our structure-preserving schemes. We respond to each major comment below and indicate the revisions we will make.

read point-by-point responses

  1. Referee: [Abstract] Abstract: the central claim that the first-order scheme exactly satisfies energy dissipation and positivity after the displacement correction (to enforce Gauss's law) and potential reconstruction (to enforce Faraday's law) is load-bearing, yet the abstract provides no indication that the energy equality or positivity bound survives the field and concentration alterations introduced by these corrections in the fully nonlinear setting. The fact that the BDF2 variant drops both energy and positivity claims suggests the corrections are not neutral with respect to those structures.

    Authors: We agree that the abstract should more clearly signal that the preservation proofs account for the post-correction fields and concentrations. The BDF2 scheme drops the energy and positivity claims because the underlying BDF2 discretization itself does not inherit the same discrete energy law or positivity bound as backward Euler, independent of the corrections; the corrections are constructed to be neutral with respect to the structures already preserved by the base scheme. In the revised manuscript we will expand the abstract to state explicitly that the proofs in Sections 3.2–3.3 establish exact preservation of all five structures after the corrections are applied. We will also add a short remark contrasting the first- and second-order cases. revision: yes

  2. Referee: [Description of the correction strategies] Description of the correction strategies: because the base discretization is constructed to preserve positivity and energy dissipation while the corrections are applied afterward, an explicit verification (or counter-example) is required to show that the altered discrete electric field and concentrations still obey the exact discrete energy law and keep concentrations non-negative; without this step the five-structure claim cannot be accepted at face value.

    Authors: We acknowledge that the manuscript would benefit from a more self-contained verification that the corrections leave the energy dissipation equality and non-negativity intact. The displacement correction is a projection onto the discrete divergence-free space that is orthogonal to the discrete gradient, and the potential reconstruction is a consistent lifting that does not alter the Slotboom variables used for positivity. Nevertheless, we agree an explicit lemma is desirable. In the revision we will insert a new lemma immediately after the description of the corrections that proves the energy law and positivity bound survive the modifications, using the algebraic properties of the corrections and the fact that they preserve the discrete inner-product structure already exploited in the base-scheme analysis. revision: yes

  3. Referee: [Numerical experiments section] Numerical experiments section: the reported tests confirm exact Gauss/Faraday preservation and convergence orders, but do not appear to include quantitative monitoring of the discrete energy dissipation rate or positivity margin after the corrections under strong nonlinear coupling; such diagnostics would be necessary to support the theoretical claims.

    Authors: The existing experiments already demonstrate exact long-time preservation of Gauss’s and Faraday’s laws and positivity of concentrations for the manufactured-solution and fixed-charge test cases. To provide direct numerical support for the energy-dissipation claim after corrections, we will augment the numerical section with additional figures that plot the discrete energy dissipation rate (computed both before and after corrections) and the minimum concentration value over time for the nonlinear ion-transport examples. These diagnostics will be added to the revised manuscript. revision: yes

Circularity Check

0 steps flagged

No significant circularity: structures preserved by explicit construction steps

full rationale

The paper constructs the scheme via Slotboom transformation for positivity, backward Euler + centered differences, then applies explicit displacement correction to enforce Gauss's law and potential reconstruction to enforce Faraday's law. The exact satisfaction of mass conservation, positivity, energy dissipation, Gauss's law, and Faraday's law follows directly from these design choices and is verified in the discrete equations. No self-citations, fitted parameters renamed as predictions, or self-definitional reductions appear; the preservation is a consequence of the algorithm definition rather than a tautological input-output loop. This is standard for structure-preserving discretizations and remains self-contained.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The central claims rest on standard numerical analysis assumptions for finite differences and time integrators plus the effectiveness of the two correction strategies; no free parameters or new entities are introduced in the abstract description.

axioms (2)
  • standard math Finite difference operators and backward Euler/BDF2 time stepping satisfy standard consistency and stability properties for the transformed system.
    Invoked implicitly for discretization and convergence claims.
  • domain assumption The correction steps can be applied without destroying positivity or energy dissipation in the nonlinear setting.
    Required for the exact preservation statements to hold.

pith-pipeline@v0.9.0 · 5488 in / 1280 out tokens · 49992 ms · 2026-05-10T17:35:44.927312+00:00 · methodology

discussion (0)

Sign in with ORCID, Apple, or X to comment. Anyone can read and Pith papers without signing in.

Reference graph

Works this paper leans on

47 extracted references · 47 canonical work pages

  1. [1]

    App lied Numerical Mathematics 223, 255–278 (2026)

    Ankur, A., Jiwari, R., Singh, S.: Finite element simulati on of modified Poisson-Nernst-Planck/Navier-Stokes model for com- pressible electrolytes under mechanical equilibrium. App lied Numerical Mathematics 223, 255–278 (2026)

    work page 2026

  2. [2]

    Physical Review E — Statistical, Nonlinear, and Soft Matter Physics 80(1), 016705 (2009)

    Baptista, M., Schmitz, R., Dünweg, B.: Simple and robust s olver for the Poisson-Boltzmann equation. Physical Review E — Statistical, Nonlinear, and Soft Matter Physics 80(1), 016705 (2009)

    work page 2009

  3. [3]

    Physi cal Review Letters 79(3), 435 (1997)

    Borukhov, I., Andelman, D., Orland, H.: Steric effects in e lectrolytes: A modified Poisson-Boltzmann equation. Physi cal Review Letters 79(3), 435 (1997)

    work page 1997

  4. [4]

    Journal of Computational Physics 501, 112791 (2024)

    Chang, C., Xin, Z., Zeng, T.: A conservative hybrid deep le arning method for Maxwell–Ampère–Nernst–Planck equation s. Journal of Computational Physics 501, 112791 (2024)

    work page 2024

  5. [5]

    Journal of Compu tational and Applied Mathematics 484, 117552 (2026)

    Chen, H., Chen, Y ., Kou, J., Sun, S.: Bound-preserving ada ptive time-stepping methods with energy stability for simu lating compressible gas flow in poroelastic media. Journal of Compu tational and Applied Mathematics 484, 117552 (2026)

    work page 2026

  6. [6]

    Physics of Particles and Nuclei L etters 17(3), 325–327 (2020)

    Chikhachev, A.: Quantum problem on the dynamics of an elec tric charge in its own field. Physics of Particles and Nuclei L etters 17(3), 325–327 (2020)

    work page 2020

  7. [7]

    SI AM Journal on Numerical Analysis 52(2), 779–807 (2014)

    Ciarlet, P ., Wu, H., Zou, J.: Edge element methods for Maxw ell’s equations with strong convergence for Gauss’ laws. SI AM Journal on Numerical Analysis 52(2), 779–807 (2014)

    work page 2014

  8. [8]

    ESAIM: Mathematical Modelling and Numerical Analysis 57(3), 1511–1551 (2023)

    Correa, C., Gatica, G., Ruiz-Baier, R.: New mixed finite el ement methods for the coupled Stokes and Poisson–Nernst–Pl anck equations in Banach spaces. ESAIM: Mathematical Modelling and Numerical Analysis 57(3), 1511–1551 (2023)

    work page 2023

  9. [9]

    Communications in Mathematical Science 21(2), 459–484 (2023)

    Ding, J., Wang, C., Zhou, S.: Convergence analysis of structure-preserving numerical methods based on Slotboom transformation for the Poisson-Nernst-Planck equations. Communications in Mathematical Science 21(2), 459–484 (2023)

    work page 2023

  10. [10]

    Entropy 23(2), 172 (2021)

    Eisenberg, R.: Maxwell equations without a polarizatio n field, Using a paradigm from biophysics. Entropy 23(2), 172 (2021)

    work page 2021

  11. [11]

    Physical Review E 90(6), 063304 (2014)

    Fahrenberger, F., Holm, C.: Computing the Coulomb inter action in inhomogeneous dielectric media via a local electr ostatics lattice algorithm. Physical Review E 90(6), 063304 (2014)

    work page 2014

  12. [12]

    The Journal of Chemical Physics 141(6) (2014)

    Fahrenberger, F., Xu, Z., Holm, C.: Simulation of electr ic double layers around charged colloids in aqueous solution of variable permittivity. The Journal of Chemical Physics 141(6) (2014)

    work page 2014

  13. [13]

    Computers and Mathematics with Applications 186, 53–83 (2025)

    Gatica, G., Inzunza, C., Ruiz-Baier, R.: Primal-mixed fi nite element methods for the coupled Biot and Poisson–Nerns t–Planck equations. Computers and Mathematics with Applications 186, 53–83 (2025)

    work page 2025

  14. [14]

    Com- munications in Mathematical Sciences 24(3), 619–644 (2026)

    Ge, Z., Xu, D.: Multiphysics finite element method for the rmo-poroelasticity with nonlinear convective transport t erm. Com- munications in Mathematical Sciences 24(3), 619–644 (2026)

    work page 2026

  15. [15]

    Journal of Scientific Computing 101(2), 51 (2024)

    Guo, Y ., Yin, Q., Zhang, Z.: A structure-preserving Impl icit Exponential Time Differencing Scheme for Maxwell–Ampè re Nernst–Planck Model. Journal of Scientific Computing 101(2), 51 (2024)

    work page 2024

  16. [16]

    Applied Mathematical Modelling 142, 115987 (2025)

    Han, X., Zhao, X., Qu, Z., Wu, Y ., Li, G.: LS-SVM-based non linear multi-physical steady-state field coupled problems computing method. Applied Mathematical Modelling 142, 115987 (2025)

    work page 2025

  17. [17]

    Advances in Computational Mathematics 48, 49 (2022) 36 Haoran Sun et al

    Hao, T., Ma, M., Xu, X.: Adaptive finite element approxima tion for steady-state Poisson-Nernst-Planck equations. Advances in Computational Mathematics 48, 49 (2022) 36 Haoran Sun et al

    work page 2022

  18. [18]

    Applied Mathema tics and Computation 287-288, 214–223 (2016)

    He, D., Pan, K.: An energy preserving finite difference sch eme for the Poisson–Nernst–Planck system. Applied Mathema tics and Computation 287-288, 214–223 (2016)

    work page 2016

  19. [19]

    Journal of Nonlinear Science 35(2), 44 (2025)

    Herda, M., Jüngel, A., Portisch, S.: Charge transport sy stems with Fermi–Dirac statistics for memristors. Journal of Nonlinear Science 35(2), 44 (2025)

    work page 2025

  20. [20]

    Applied Mathematics and Comput ation 524, 130029 (2026)

    Hoang, T., Ehrhardt, M.: A generalized second-order pos itivity-preserving numerical method for non-autonomous d ynamical systems with applications. Applied Mathematics and Comput ation 524, 130029 (2026)

    work page 2026

  21. [21]

    Biophysical Journal 116(2), 270–282 (2019)

    Horng, T., Eisenberg, R., Liu, C., Bezanilla, F.: Contin uum gating current models computed with consistent interac tions. Biophysical Journal 116(2), 270–282 (2019)

    work page 2019

  22. [22]

    BIT Numerical Mathematics 66(2), 19 (2026)

    Hu, X., Dai, X., Xiao, A.: Weak convergence rate of positi vity-preserving truncated Euler–Maruyama method for mult i- dimensional stochastic differential equations with positi ve solutions. BIT Numerical Mathematics 66(2), 19 (2026)

    work page 2026

  23. [23]

    SIAM Journal on Numerical Analysis 55(6), 2787–2810 (2017)

    Irwin, Y ., Jun, Z.: Edge Element Method for Optimal Contr ol of Stationary Maxwell System with Gauss Law. SIAM Journal on Numerical Analysis 55(6), 2787–2810 (2017)

    work page 2017

  24. [24]

    Mathe- matical Methods in the Applied Sciences 39(5), 1183–1196 (2016)

    Lee, M., Ko, M., Kim, Y .: Orthotropic conductivity recon struction with virtual-resistive network and Faraday’s la w. Mathe- matical Methods in the Applied Sciences 39(5), 1183–1196 (2016)

    work page 2016

  25. [25]

    CSIAM Transactions on Applied Mathematics (2026 )

    Li, J., Wang, K., Ju, L.: Linear Maximum Bound Principle P reserving Finite Difference Schemes for the Convective Alle n-Cahn Equation. CSIAM Transactions on Applied Mathematics (2026 )

    work page 2026

  26. [26]

    Mathematics and Computers in Simulation 239, 172–191 (2026)

    Li, M., Zou, G., Wang, B.: A new fully-decoupled energy-s table BDF2-FEM scheme for the electro-hydrodynamic equati ons. Mathematics and Computers in Simulation 239, 172–191 (2026)

    work page 2026

  27. [27]

    Communications in Nonlinear Science and Numer ical Simulation 159, 109826 (2026)

    Li, Y ., Wang, Y ., Tan, S., Ni, G.: A temporally second-ord er positivity-preserving unified gas-kinetic scheme for pl asma simulation. Communications in Nonlinear Science and Numer ical Simulation 159, 109826 (2026)

    work page 2026

  28. [28]

    Mathema tics and Computers in Simulation 244, 162–180 (2026)

    Lin, X., He, M., Sun, P .: Optimal convergence arbitrary L agrangian–Eulerian finite element methods in energy norm fo r Poisson–Nernst–Planck moving boundary problems. Mathema tics and Computers in Simulation 244, 162–180 (2026)

    work page 2026

  29. [29]

    S IAM Journal on Applied Mathematics 78(1), 226–245 (2018)

    Liu, P ., Ji, X., Xu, Z.: Modified Poisson-Nernst-Planck m odel with accurate Coulomb correlation in variable media. S IAM Journal on Applied Mathematics 78(1), 226–245 (2018)

    work page 2018

  30. [30]

    Computers and Mathematics with Applications 102, 95–112 (2021)

    Liu, Y ., Shu, S., Wei, H., Y ang, Y .: A virtual element method for the steady-state Poisson-Nernst-Planck equations on polygonal meshes. Computers and Mathematics with Applications 102, 95–112 (2021)

    work page 2021

  31. [31]

    : Variationally and numerically coupled water-wave and sur f zone hydrodynamics

    Lu, Y ., Gidel, F., Bunnik, T., Bokhove, O., Kelmanson, M. : Variationally and numerically coupled water-wave and sur f zone hydrodynamics. Journal of Engineering Mathematics 157(1): 2 (2026)

    work page 2026

  32. [32]

    Communications in Nonline ar Science and Numerical Simulation 159, 109915 (2026)

    Luo, F., Tang, Q.: T wo new local structure-preserving sc hemes for the GBBM-KdV equation. Communications in Nonline ar Science and Numerical Simulation 159, 109915 (2026)

    work page 2026

  33. [33]

    SIAM Jou rnal on Applied Mathematics 81(4), 1645–1667 (2021)

    Ma, M., Xu, Z., Zhang, L.: Modified Poisson-Nernst-Planc k model with Coulomb and hard-sphere correlations. SIAM Jou rnal on Applied Mathematics 81(4), 1645–1667 (2021)

    work page 2021

  34. [34]

    Communications in Nonlinear Science and Numerical Simulat ion 132, 107933 (2024)

    Nastasi, G., Borzì, A., Romano, V .: Optimal control of a s emiclassical Boltzmann equation for charge transport in gr aphene. Communications in Nonlinear Science and Numerical Simulat ion 132, 107933 (2024)

    work page 2024

  35. [35]

    Compu ters and Fluids 193, 104279 (2019)

    Pimenta, F., Alves, M.: A coupled finite-volume solver fo r numerical simulation of electrically-driven flows. Compu ters and Fluids 193, 104279 (2019)

    work page 2019

  36. [36]

    SMAI-Journal of computational mathema tics 3, 53–89 (2017)

    Pinto, M., Sonnendrücker, E.: Compatible Maxwell solve rs with particles i: conforming and non-conforming 2d schem es with a strong Ampere law. SMAI-Journal of computational mathema tics 3, 53–89 (2017)

    work page 2017

  37. [37]

    SMAI-Journal of computational mathem atics 3, 91–116 (2017)

    Pinto, M., Sonnendrücker, E.: Compatible Maxwell solve rs with particles ii: conforming and non-conforming 2d sche mes with a strong Faraday law. SMAI-Journal of computational mathem atics 3, 91–116 (2017)

    work page 2017

  38. [38]

    J ournal of Computational and Applied Mathematics 443, 115759 (2024)

    Qiao, T., Qiao, Z., Sun, S.: An unconditionally energy st able linear scheme for Poisson–Nernst–Planck equations. J ournal of Computational and Applied Mathematics 443, 115759 (2024)

    work page 2024

  39. [39]

    SIAM Journal on Applied Mathematics 83(2), 374–393 (2023)

    Qiao, Z., Xu, Z., Yin, Q., Zhou, S.: A Maxwell–Ampère Nern st–Planck framework for modeling charge dynamics. SIAM Journal on Applied Mathematics 83(2), 374–393 (2023)

    work page 2023

  40. [40]

    Journal of Computational Physics 475, 111845 (2023)

    Qiao, Z., Xu, Z., Yin, Q., Zhou, S.: Structure-preservin g numerical method for Maxwell-Ampère Nernst-Planck model. Journal of Computational Physics 475, 111845 (2023)

    work page 2023

  41. [41]

    IMA Journal of N umerical Analysis (2025)

    Qin, Y ., Wang, C.: A second-order accurate, positivity- preserving numerical scheme for the Pois- son–Nernst–Planck–Navier–Stokes system. IMA Journal of N umerical Analysis (2025). Draf027

    work page 2025

  42. [42]

    Journal of Mathematical Physics 44(6), 2521–2533 (2003)

    Salmela, A.: An algebraic method for solving the SU(3) Ga uss law. Journal of Mathematical Physics 44(6), 2521–2533 (2003)

    work page 2003

  43. [43]

    SIAM Journal on Numerical Analysis 62(4), 2004–2024 (2024)

    Tong, F., Cai, Y .: Positivity preserving and mass conser vative projection method for the Poisson–Nernst–Planck eq uation. SIAM Journal on Numerical Analysis 62(4), 2004–2024 (2024)

    work page 2004

  44. [44]

    Applied Numerical Mathematics 163, 204–220 (2021)

    Wan, J., Dong, W., Zhao, Y ., Song, S.: Second-order two-s cale analysis for heating effect of periodical composite str ucture. Applied Numerical Mathematics 163, 204–220 (2021)

    work page 2021

  45. [45]

    Axioms 15(2), 126 (2026)

    Yuan, M., Liu, J., Ma, P ., Li, M.: An Energy-Stable S-SA V F inite Element Method for the Generalized Poisson-Nernst-P lanck Equation. Axioms 15(2), 126 (2026)

    work page 2026

  46. [46]

    Physical Review E — Statistical, Nonlinear, and Soft Matter Physics 84(2), 021901 (2011)

    Zhou, S., Wang, Z., Li, B.: Mean-field description of ioni c size effects with nonuniform ionic sizes: A numerical appro ach. Physical Review E — Statistical, Nonlinear, and Soft Matter Physics 84(2), 021901 (2011)

    work page 2011

  47. [47]

    Journal of Computational and Applied Mat hematics 482, 117284 (2026)

    Zhu, W., Ji, G.: A posteriori error estimates of the weak G alerkin finite element method for time-dependent Poisson-N ernst- Planck equations. Journal of Computational and Applied Mat hematics 482, 117284 (2026)

    work page 2026