Positivity Preserving Hybridizable Discontinuous Galerkin Scheme for Solving PNP Model

This dissertation focuses on deriving and implementing a Hybridizable Discontinuous Galerkin (HDG) scheme for solving the Poisson-Nernst-Planck (PNP) problem, a type of partial differential equation (PDE). The PNP problem is a system of equations that models the movement of charged particles in fluids.

The motivation for this study hinges on the importance of proper ion transport phenomena. Ion transportation exists in many realms, including semiconductors, electrochemistry, and biological systems {Flavell2014, Flavell2017, Horng2012}. In biological systems, ions move through ion channels, which operate as a gate. The proper function of ion channels are responsible for the health and well-being of humans and animals. A breakdown of these ion channels can lead to a multitude of recorded illnesses: Brugada syndrome, QT syndrome, etc {Kim2014, Konrad2021, Marban2002}. The study of how ions move through an ion channel gives qualitative reasoning that can be used to explain this important biological phenomena.

We use the log-density formulation of the PNP system, as proposed in {Metti2016}, to model the concentration of two charged particles through an ion channel in both one-dimension and two-dimensions. Using the log-density formulation is what ensures the density of our charged particles preserves positive, i.e., c_i = exp(u_i) and c_i > 0. To this formulation, we implement a spatial HDG scheme in combination with an implicit Euler temporal discretization. To obtain a higher convergence order in time, we later use a Crank-Nicolson temporal discretization.

This is a computationally expensive scheme that can become even more convoluted with increasing the number of charged particles modeled. One way that we aim to reduce computational cost is by incorporating a simple adaptive time stepping algorithm that utilized two half-steps of the implicit Euler method. Adaptive time stepping is a great technique to advance the solution through discontinuities while also reducing the computational cost. We conclude our work by providing several avenues that can be taken for future research.