  A second order accurate (in time) numerical scheme is proposed and analyzed
for the Poisson-Nernst-Planck equation (PNP) system, reformulated as a
non-constant mobility $H^{-1}$ gradient flow in the Energetic Variational
Approach (EnVarA). The centered finite difference is taken as the spatial
discretization. Meanwhile, the highly nonlinear and singular nature of the
logarithmic energy potentials has always been the essential difficulty to
design a second order accurate scheme in time, while preserving the variational
energetic structures. The mobility function is updated with a second order
accurate extrapolation formula, for the sake of unique solvability. A modified
Crank-Nicolson scheme is used to approximate the logarithmic term, so that its
inner product with the discrete temporal derivative exactly gives the
corresponding nonlinear energy difference; henceforth the energy stability is
ensured for the logarithmic part. In addition, nonlinear artificial
regularization terms are added in the numerical scheme, so that the
positivity-preserving property could be theoretically proved, with the help of
the singularity associated with the logarithmic function. Furthermore, an
optimal rate convergence analysis is provided in this paper, in which the
higher order asymptotic expansion for the numerical solution, the rough error
estimate and refined error estimate techniques have to be included to
accomplish such an analysis. This work combines the following theoretical
properties for a second order accurate numerical scheme for the PNP system: (i)
second order accuracy in both time and space, (ii) unique solvability and
positivity, (iii) energy stability, and (iv) optimal rate convergence. A few
numerical results are also presented.
