A conservative finite difference scheme for Poisson-Nernst-Planck equations

A macroscopic model to describe the dynamics of ion transport in ion channels is the Poisson-Nernst-Planck (PNP) equations. In this paper, we develop a finite-difference method for solving PNP equations, second-order accurate in both space and time. We use the physical parameters specifically suited toward the modeling of ion channels. We present a simple iterative scheme to solve the system of nonlinear equations resulting from discretizing the equations implicitly in time, which is demonstrated to converge in a few iterations. We place emphasis on ensuring numerical methods to have the same physical properties that the PNP equations themselves also possess, namely conservation of total ions, correct rates of energy dissipation, and positivity of the ion concentrations. We describe in detail an approach to derive a finite-difference method that preserves the total concentration of ions exactly in time. In addition, we find a set of sufficient conditions on the step sizes of the numerical method that assure positivity of the ion concentrations. Further, we illustrate that, using realistic values of the physical parameters, the conservation property is critical in obtaining correct numerical solutions over long time scales.