PHYSICS-PRESERVING FINITE DIFFERENCE SCHEMES FOR THE POISSON-NERNST-PLANCK EQUATIONS

The Poisson-Nernst-Planck equations are a system of nonlinear di erential equations that describe ow of charged particles in solution. This dissertation is about the design of numerical schemes to solve this system which preserves global properties exhibited by the system. There are two major advances presented. The rst is the design of schemes that conserve mass globally when the system is coupled with no- ux boundary conditions. Most notably, a scheme using central di erencing and TR-BDF2 achieves second order accuracy in both space and time, while also conserving global mass is presented. The second is the design of a more general scheme that preserves the time-varying properties of the free energy of the system. One such a scheme uses central di erencing in space and trapezoidal integration in time to achieve second order accuracy in both space and time, while also preserving the energy dynamics, but at the cost of requiring positivity of the solution. There is also a discussion of solution methods: the classic Newton iteration scheme is compared with a modi ed Gummel iteration scheme for the purpose of solving the transient equations. The intended application of this work is the modeling of ion channels, and many of the simulations presented use parameters consistent with models of ion channels.Ph.D. in Applied Mathematics, July 201