  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.

Numerical Analysis;Numerical Analysis