Adaptive Time-stepping Schemes 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 number of fields. In this article, we develop an adaptive time-stepping scheme for the solution of the PNP-FBV equations based on two time-stepping methods: a fully implicit (BDF2) method, and an implicit-explicit (SBDF2) method. We present simulations under both current and voltage boundary conditions and demonstrate the ability to simulate a large range of parameters, including any value of the singular perturbation parameter $\epsilon$. When the underlying dynamics is one that would have the solutions converge to a steady-state solution, we observe that the adaptive time-stepper based on the SBDF2 method produces solutions that ``nearly'' converge to the steady state and that, simultaneously, the time-step sizes stabilize to a limiting size $dt_\infty$. In the companion to this article \cite{YPD_Part2}, we linearize the SBDF2 scheme about the steady-state solution and demonstrate that the linearized scheme is conditionally stable. This conditional stability is the cause of the adaptive time-stepper's behaviour. While the adaptive time-stepper based on the fully-implicit (BDF2) method is not subject to such time-step constraints, the required nonlinear solve yields run times that are significantly longer.