An Unconditionally Energy Stable Linear Scheme for Poisson–Nernst–Planck Equations

Tian Qiao, Zhonghua Qiao, Shuyu Sun, Shenggao Zhou

Research output: Journal article publicationJournal articleAcademic researchpeer-review

1 Citation (Scopus)


This paper proposes a linear, unconditionally energy-stable scheme for the Poisson–Nernst–Planck (PNP) equations. Based on a gradient-flow formulation of the PNP equations, the energy factorization approach is applied to linearize the logarithm function at the previous time step, resulting in a linear semi-implicit scheme. Numerical analysis is conducted to illustrate that the proposed fully discrete scheme has desired properties at a discrete level, such as unconditional unique solvability, mass conservation, and energy dissipation. Numerical simulations verify that the proposed scheme, as expected, is first-order accurate in time and second-order accurate in space. Further numerical tests confirm that the proposed scheme can indeed preserve the desired properties. Applications of our numerical scheme to the simulations of electrolyte solutions demonstrate that, as a linear energy stable scheme of efficiency, it will be promising in simulating complicated transport phenomena of charged systems.

Original languageEnglish
Article number115759
JournalJournal of Computational and Applied Mathematics
Publication statusPublished - Jun 2024


  • Electric double layer
  • Energy stability
  • Mass conservation
  • Poisson–Nernst–Planck equations

ASJC Scopus subject areas

  • Computational Mathematics
  • Applied Mathematics


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