  In this paper, we design, analyze, and numerically validate positive and
energy-dissipating schemes for solving the time-dependent multi-dimensional
system of Poisson-Nernst-Planck (PNP) equations, which has found much use in
the modeling of biological membrane channels and semiconductor devices. The
semi-implicit time discretization based on a reformulation of the system gives
a well-posed elliptic system, which is shown to preserve solution positivity
for arbitrary time steps. The first order (in time) fully-discrete scheme is
shown to preserve solution positivity and mass conservation unconditionally,
and energy dissipation with only a mild $O(1)$ time step restriction. The
scheme is also shown to preserve the steady-state. For the fully second order
(in both time and space) scheme with large time steps, solution positivity is
restored by a local scaling limiter, which is shown to maintain the spatial
accuracy. These schemes are easy to implement. Several three-dimensional
numerical examples verify our theoretical findings and demonstrate the
accuracy, efficiency, and robustness of the proposed schemes, as well as the
fast approach to steady states.
