Positive-preserving, mass conservative linear schemes for the Possion-Nernst-Planck equations

TL;DR

This paper introduces the first second-order linear schemes for the Poisson-Nernst-Planck equations that preserve positivity and mass conservation, with proven energy stability and validated by extensive numerical tests.

Contribution

It develops the first second-order exponential time differencing schemes that are linear, positivity-preserving, and mass conservative for PNP equations, overcoming previous limitations.

Findings

Schemes preserve positivity and mass conservation at full discretization.

Second-order scheme dissipates the modified energy.

Numerical results confirm theoretical properties and performance.

Abstract

The first-order linear positivity preserving schemes in time are available for the time dependent Poisson-Nernst-Planck (PNP) equations, second-order linear ones are still challenging. In this paper, we propose the first- and second-order exponential time differencing schemes with the finite difference spatial discretization for PNP equations, based on the Slotboom transformation of the Nernst-Planck equation. The proposed schemes are linear and preserve the mass conservation and positivity preservation of ion concentration at full discrete level without any constraints on the time step size. The corresponding energy stability analysis is also presented, demonstrating that the second-order scheme can dissipate the modified energy. Extensive numerical results are carried out to support the theoretical findings and showcase the performance of the proposed schemes.