Unconditional positivity-preserving and energy stable schemes for a reduced Poisson-Nernst-Planck system

TL;DR

This paper introduces a second order, positivity-preserving, and energy stable numerical scheme for a reduced Poisson-Nernst-Planck system, effectively modeling ionic channels with proven stability and efficiency.

Contribution

The paper develops a novel second order scheme that guarantees positivity and energy stability for the reduced PNP system, with rigorous analysis and numerical validation.

Findings

Scheme is positivity-preserving regardless of time step size

Numerical results confirm accuracy and efficiency

Fast convergence to steady states demonstrated

Abstract

The Poisson-Nernst-Planck (PNP) system is a widely accepted model for simulation of ionic channels. In this paper, we design, analyze, and numerically validate a second order unconditional positivity-preserving scheme for solving a reduced PNP system, which can well approximate the three dimensional ion channel problem. Positivity of numerical solutions is proven to hold true independent of the size of time steps and the choice of the Poisson solver. The scheme is easy to implement without resorting to any iteration method. Several numerical examples further confirm the positivity-preserving property, and demonstrate the accuracy, efficiency, and robustness of the proposed scheme, as well as the fast approach to steady states.