Communications in Mathematical Sciences

Volume 21 (2023)

Number 2

Convergence analysis of structure-preserving numerical methods based on Slotboom transformation for the Poisson–Nernst–Planck equations

Pages: 459 – 484



Jie Ding (School of Science, Jiangnan University, Wuxi, Jiangsu, China)

Cheng Wang (Department of Mathematics, University of Massachusetts, North Dartmouth, Mass., U.S.A.)

Shenggao Zhou (School of Mathematical Sciences, MOE-LSC, and CMA-Shanghai, Shanghai Jiao Tong University, Shanghai, China)


The analysis of structure-preserving numerical methods for the Poisson–Nernst–Planck (PNP) system has attracted growing interests in recent years. A class of numerical algorithms have been developed based on the Slotboom reformulation, and the mass conservation, ionic concentration positivity, free-energy dissipation have been proved at a discrete level. Nonetheless, a rigorous convergence analysis for these Slotboom reformulation-based, structure-preserving schemes has been an open problem for a long time. In this work, we provide an optimal rate convergence analysis and error estimate for finite difference schemes based on the Slotboom reformulation. Different options of mobility average at the staggered mesh points are considered in the finite-difference spatial discretization, such as the harmonic mean, geometric mean, arithmetic mean, and entropic mean. A semi-implicit temporal discretization is applied, which in turn results in a non-constant coefficient, positive-definite linear system at each time step. A higher order asymptotic expansion is applied in the consistency analysis, and such a higher order consistency estimate is necessary to control the discrete maximum norm of the concentration variables. In convergence estimate, the harmonic mean for the mobility average, which turns out to bring lots of convenience in the theoretical analysis, is taken for simplicity, while other options of mobility average would also lead to the desired error estimate, with more technical details involved. As a result, an optimal rate convergence analysis on concentrations, electric potential, and ionic fluxes is derived, which is the first such result for the structure-preserving numerical schemes based on the Slotboom reformulation. It is remarked that the convergence analysis leads to a theoretical justification of the conditional energy dissipation analysis, which relies on the maximum norm bounds of the concentration and the gradient of the electric potential. Some numerical results are also presented to demonstrate the accuracy and structure-preserving performance of the associated schemes.


Poisson–Nernst–Planck equations, Slotboom reformulation, mobility average, convergence analysis and error estimate, higher order consistency estimate

2010 Mathematics Subject Classification

35K55, 35K61, 65M06, 65M12

Received 29 December 2021

Received revised 21 May 2022

Accepted 6 June 2022

Published 1 February 2023