# Positivity-Preserving Hybridizable Discontinuous Galerkin Scheme for Solving PNP Model

## TL;DR

This paper introduces a new numerical method for solving equations that model charged particle transport while ensuring physical accuracy.

## Contribution

A positivity-preserving HDG scheme for PNP equations with energy stability and mass conservation proofs is proposed.

## Key findings

- The HDG scheme ensures positivity of charged particle densities using a log-density formulation.
- The proposed method is proven to be energy stable and mass conserving.
- Numerical simulations confirm the accuracy of the scheme in one and two spatial dimensions.

## Abstract

We introduce a hybridizable discontinuous Galerkin (HDG) scheme for solving the Poisson–Nernst–Planck (PNP) equations. The log-density formulation as introduced by Metti et al. in their paper “Energetically stable discretizations for charge transport and electrokinetic models. J. Comput. Phys. 2016, 306, 1-18” is utilized to ensure the positivity of the densities of the charged particles. We further prove that our fully discrete scheme is energy stable and mass conserving. Numerical simulations are provided to demonstrate the accuracy of the scheme in one and two spatial dimensions. A derivation of an HDG-DG space–time scheme is given, with implementation and convergence analysis left to future work.