A dynamic mass transport method for Poisson-Nernst-Planck equations

TL;DR

This paper introduces a novel dynamic mass transport method for solving Poisson-Nernst-Planck equations that guarantees physical properties like positivity, mass conservation, and energy dissipation through a variational scheme.

Contribution

The paper develops a semi-discrete and fully discrete variational scheme for PNP equations that inherently preserves key physical properties regardless of discretization parameters.

Findings

The scheme enforces positivity and energy law naturally.

Numerical experiments confirm convergence and structure preservation.

Method is applicable for accurate and stable simulations of PNP systems.

Abstract

A dynamic mass-transport method is proposed for approximately solving the Poisson-Nernst-Planck(PNP) equations. The semi-discrete scheme based on the JKO type variational formulation naturally enforces solution positivity and the energy law as for the continuous PNP system. The fully discrete scheme is further formulated as a constrained minimization problem, shown to be solvable, and satisfy all three solution properties (mass conservation, positivity and energy dissipation) independent of time step size or the spatial mesh size. Numerical experiments are conducted to validate convergence of the computed solutions and verify the structure preserving property of the proposed scheme.