A Conservative Finite Difference Scheme for Poisson-Nernst-Planck Equations

TL;DR

This paper introduces a second-order accurate finite difference scheme for solving Poisson-Nernst-Planck equations, emphasizing physical property preservation like ion conservation and energy dissipation, crucial for long-term simulations.

Contribution

The paper develops a finite difference method that preserves key physical properties of PNP equations and demonstrates its effectiveness with realistic ion channel parameters.

Findings

Method achieves second-order accuracy in space and time.

Conservation of total ions is maintained exactly in the numerical scheme.

The scheme converges rapidly with a few iterations.

Abstract

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, which is second-order accurate in both space and time. We use the physical parameters specifically suited toward the modelling 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 and correct rates of energy dissipation. We describe in detail an approach to derive a finite-difference method that preserves the total concentration of ions exactly in time. 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.