A structure-preserving implicit exponential time differencing scheme for Maxwell-Amp`ere Nernst-Planck model

TL;DR

This paper introduces a novel structure-preserving numerical scheme for the Maxwell-Ampere Nernst-Planck model that ensures positivity, energy dissipation, and Gauss's law adherence, validated through numerical experiments.

Contribution

It develops a decoupled implicit exponential time differencing scheme with positivity preservation and energy stability for the MANP model, incorporating a curl-free relaxation algorithm.

Findings

The scheme preserves positivity of concentrations.

It satisfies the energy dissipation law.

Numerical experiments confirm accuracy and structure preservation.

Abstract

The transport of charged particles, which can be described by the Maxwell-Ampere Nernst-Planck (MANP) framework, is essential in various applications including ion channels and semiconductors. We propose a decoupled structure-preserving numerical scheme for the MANP model in this work. The Nernst-Planck equations are treated by the implicit exponential time differencing method associated with the Slotboom transform to preserve the positivity of the concentrations. In order to be effective with the Fast Fourier Transform, additional diffusive terms are introduced into Nernst-Planck equations. Meanwhile, the correction is introduced in the Maxwell-Ampere equation to fulfill Gauss's law. The curl-free condition for electric displacement is realized by a local curl-free relaxation algorithm whose complexity is O(N). We present sufficient restrictions on the time and spatial steps to satisfy the positivity and energy dissipation law at a discrete level. Numerical experiments are conducted to validate the expected numerical accuracy and demonstrate the structure-preserving properties of the proposed method.