Local Structure-Preserving Relaxation Method for Charged Systems on Unstructured Meshes

TL;DR

This paper introduces a local relaxation method for solving modified Poisson--Nernst--Planck equations on unstructured meshes, effectively handling sharp boundary layers and ensuring physical constraints like positivity and free-energy decrease.

Contribution

It presents a novel relaxation algorithm that respects Gauss's law and guarantees positivity and energy dissipation for charged systems on unstructured meshes.

Findings

Method guarantees positivity of ionic and solvent concentrations.

Free energy decreases during iterative updates.

Accurate and robust simulation of sharp boundary layers.

Abstract

This work considers charged systems described by the modified Poisson--Nernst--Planck (PNP) equations, which incorporate ionic steric effects and the Born solvation energy for dielectric inhomogeneity. Solving the steady-state modified PNP equations poses numerical challenges due to the emergence of sharp boundary layers caused by small Debye lengths, particularly when local ionic concentrations reach saturation. To address this, we first reformulate the steady-state problem as a constraint optimization, where the ionic concentrations on unstructured Delaunay nodes are treated as fractional particles moving along edges between nodes. The electric fields are then updated to minimize the objective free energy while satisfying the discrete Gauss's law. We develop a local relaxation method on unstructured meshes that inherently respects the discrete Gauss's law, ensuring curl-free electric fields. Numerical analysis demonstrates that the optimal mass of the moving fractional particles guarantees the positivity of both ionic and solvent concentrations. Additionally, the free energy of the charged system consistently decreases during successive updates of ionic concentrations and electric fields. We conduct numerical tests to validate the expected numerical accuracy, positivity, free-energy dissipation, and robustness of our method in simulating charged systems with sharp boundary layers.