Convergence Analysis of a Symmetrical and Positivity-Preserving Finite Difference Scheme for 1D Poisson–Nernst–Planck System

Abstract

The Poisson–Nernst–Planck (PNP) system is a nonlinear coupled system that describes the motion of ionic particles. As the exact solution of the system is not available, numerical investigations are essentially important, and there are quite a lot of numerical methods proposed in the existing literature. However, the theoretical analysis is usually neglected due to the complicated nature of the PNP system. In this paper, a theoretical investigation for a symmetrical finite difference method proposed in the previous literature was conducted. An L2 error estimate of O(τ+h2) was derived for the numerical scheme in 1D, where τ denotes the time step size and h denotes the spatial mesh size, respectively. Numerical results confirm the theoretical analysis. More importantly, a positivity-preserving condition for the scheme is provided with rigorously theoretical justification.