A free energy satisfying discontinuous Galerkin method for one-dimensional Poisson--Nernst--Planck systems

TL;DR

This paper introduces a high-order discontinuous Galerkin method for solving time-dependent Poisson-Nernst-Planck systems that preserves free energy dissipation and positivity, ensuring accurate and stable simulations.

Contribution

It develops an arbitrary-order DG method that satisfies discrete free energy dissipation and positivity preservation for Poisson-Nernst-Planck equations.

Findings

The method satisfies discrete free energy dissipation law.

Numerical solutions maintain positivity and mass conservation.

High resolution and stability demonstrated through numerical examples.

Abstract

We design an arbitrary-order free energy satisfying discontinuous Galerkin (DG) method for solving time-dependent Poisson-Nernst-Planck systems. Both the semi-discrete and fully discrete DG methods are shown to satisfy the corresponding discrete free energy dissipation law for positive numerical solutions. Positivities of numerical solutions are enforced by an accuracy-preserving limiter in reference to positive cell averages. Numerical examples are presented to demonstrate the high resolution of the numerical algorithm and to illustrate the proven properties of mass conservation, free energy dissipation, as well as the preservation of steady states.