A free energy satisfying finite difference method for Poisson--Nernst--Planck equations

TL;DR

This paper introduces a second-order finite difference method for Poisson-Nernst-Planck equations that conserves mass, preserves positivity, and satisfies a discrete free energy dissipation law, ensuring stable and physically meaningful solutions.

Contribution

The paper presents a novel finite difference scheme that guarantees mass conservation, positivity, and free energy dissipation for Poisson-Nernst-Planck equations, with proven stability and equilibrium preservation.

Findings

Method achieves second-order accuracy in space.

Numerical results confirm mass conservation and positivity.

Scheme demonstrates energy stability and equilibrium preservation.

Abstract

In this work we design and analyze a free energy satisfying finite difference method for solving Poisson-Nernst-Planck equations in a bounded domain. The algorithm is of second order in space, with numerical solutions satisfying all three desired properties: i) mass conservation, ii) positivity preserving, and iii) free energy satisfying in the sense that these schemes satisfy a discrete free energy dissipation inequality. These ensure that the computed solution is a probability density, and the schemes are energy stable and preserve the equilibrium solutions. Both one and two-dimensional numerical results are provided to demonstrate the good qualities of the algorithm, as well as effects of relative size of the data given.