A Variable Step Size Implicit-Explicit Scheme for the Solution of the Poisson-Nernst-Planck Equations

The Poisson-Nernst-Planck equations with generalized Frumkin-Butler-Volmer boundary conditions (PNP-FBV) describe ion transport with Faradaic reactions and have applications in a wide variety of fields. In this article, we develop a variable step size implicit-explicit time stepping scheme for the solution of the PNP-FBV equations. We test our numerical scheme on a simplified "toy" version of the PNP-FBV equations, paying special care to the treatment of the coupled nonlinear terms in the boundary condition. We evaluate various ways of incorporating the boundary condition into the scheme and a method based on ghost points is chosen for its favorable numerical properties compared to the alternatives. In fact, we observe that when the underlying dynamics is one that would have the solutions converge to a steady state solution, the numerical simulation does not result in the time-step sizes growing larger along with the expected convergence. We observe that the time-step sizes threshold at a particular step size. By performing an A-stability analysis we demonstrate that this thresholding does not appear to be due to a stability constraint. Using the developed numerical method, we are able to run simulations with a large range of parameters, including any value of the singular perturbation parameter epsilon.