Positivity-Preserving Hybridizable Discontinuous Galerkin Scheme for Solving PNP Model

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.