Five-Structures Preserving Algorithm for charge dynamics model

TL;DR

This paper introduces fast, structure-preserving numerical algorithms for nonlinear Maxwell-Ampere Nernst-Planck equations, ensuring key physical laws and properties are exactly maintained during simulations.

Contribution

It develops first- and second-order schemes that exactly preserve mass, positivity, energy dissipation, and electromagnetic laws, with proven error estimates and validated through numerical experiments.

Findings

Schemes exactly conserve mass, Gauss's law, and Faraday's law.

Numerical experiments confirm convergence and positivity.

Simulations demonstrate long-term preservation of physical laws.

Abstract

This paper develops a family of fast, structure-preserving numerical algorithms for the nonlinear Maxwell-Ampere Nernst-Planck equations. For the first-order scheme, the Slotboom transformation rewrites the Nernst-Planck equation to enable positivity preservation. The backward Euler method and centered finite differences discretize the transformed system. Two correction strategies are introduced: one enforces Gauss's law via a displacement correction, and the other preserves Faraday's law through potential reconstruction. The fully discrete scheme exactly satisfies mass conservation, concentration positivity, energy dissipation, Gauss's law, and Faraday's law, with established error estimates. The second-order scheme adopts BDF2 time discretization while retaining the same structure-preserving strategies, exactly conserving mass, Gauss's law, and Faraday's law. Numerical experiments validate both schemes using analytical solutions, confirming convergence orders and positivity preservation. Simulations of ion transport with fixed charges demonstrate exact preservation of Gauss's and Faraday's laws over long-time evolution, reproducing electrostatic attraction, ion accumulation, and electric field screening. The results fully support the theoretical analysis and the schemes' stability and superior performance.