Positivity preserving finite difference methods for Poisson-Nernst-Planck equations with steric interactions: Application to slit-shaped nanopore conductance

To study ion transport in electrolyte solutions, we propose numerical methods for a modified Poisson-Nernst-Planck model with ionic steric effects (SPNP). Positivity preserving schemes based on harmonic-mean approximations are proposed on a non-uniform mesh for the spatial discretization of the SPNP equations. Both explicit and semi-implicit discretization are considered in time. Numerical analysis shows that explicit forward Euler and semi-implicit trapezoidal discretization lead to schemes that maintain fully discrete solution positivity under a constraint on a mesh ratio, while the semi-implicit backward Euler discretization maintain fully discrete solution positivity unconditionally. We further study the condition numbers of the matrix associated with the semi-implicit backward Euler discretization, and establish an upper bound on condition numbers, indicating that the developed discretization based on harmonic-mean approximations effectively solves the issue of the presence of large condition numbers when using the Slotboom-type variables. Further numerical simulations confirm the analysis results on accuracy, positivity, and bounded condition numbers. The proposed schemes are also applied to study practical applications, such as the impact of self and cross steric interactions on ion distribution and rectifying behavior in a slit-shaped nanopore with surface charges. Possible extensions of the numerical methods to other modified PNP models with ion correlations and steric effects are also discussed. (C) 2019 Elsevier Inc. All rights reserved.