Positivity Preserving Hybridizable Discontinuous Galerkin Scheme for Solving PNP Model

<p>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.</p><p>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.</p><p>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., <i>c</i><i>_i</i> = exp(<i>u</i><i>_i</i>) and <i>c</i><i>_i</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.</p><p>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.</p>