An energy preserving finite difference scheme for the Poisson-Nernst-Planck system

TL;DR

This paper introduces a semi-implicit finite difference scheme for the Poisson-Nernst-Planck system that conserves mass and energy, is computationally efficient, and achieves second-order spatial and first-order temporal accuracy.

Contribution

It presents a linear-solve finite difference method for the nonlinear Poisson-Nernst-Planck system with proven energy decay and mass conservation, and demonstrates its convergence and extendability.

Findings

Method is second order in space

Method is first order in time

Efficient linear system solution at each step

Abstract

In this paper, we construct a semi-implicit finite difference method for the time dependent Poisson-Nernst-Planck system. Although the Poisson-Nernst-Planck system is a nonlinear system, the numerical method presented in this paper only needs to solve a linear system at each time step, which can be done very efficiently. The rigorous proof for the mass conservation and electric potential energy decay are shown. Moreover, mesh refinement analysis shows that the method is second order convergent in space and first order convergent in time. Finally we point out that our method can be easily extended to the case of multi-ions.