  The analysis of structure-preserving numerical methods for the
Poisson--Nernst--Planck (PNP) system has attracted growing interests in recent
years. 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 results 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.
