  In this paper, we construct a semi-implicit finite difference method for the
time dependent Poisson-Nernst-Planck system. Although the Poisson-Nernst-Planck
system is a nonlinear system, the numerical method presented in this paper only
needs to solve a linear system at each time step, which can be done very
efficiently. The rigorous proof for the mass conservation and electric
potential energy decay are shown. Moreover, mesh refinement analysis shows that
the method is second order convergent in space and first order convergent in
time. Finally we point out that our method can be easily extended to the case
of multi-ions.
