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The Poisson-Nernst-Planck equations with generalized Frumkin-Butler-Volmerboundary conditions (PNP-FBV) describe ion transport with Faradaic reactions,and have applications in a number of fields. In this article, we develop anadaptive time-stepping scheme for the solution of the PNP-FBV equations basedon two time-stepping methods: a fully implicit (BDF2) method, and animplicit-explicit (SBDF2) method. We present simulations under both current andvoltage boundary conditions and demonstrate the ability to simulate a largerange of parameters, including any value of the singular perturbation parameter$\epsilon$. When the underlying dynamics is one that would have the solutionsconverge to a steady-state solution, we observe that the adaptive time-stepperbased on the SBDF2 method produces solutions that ``nearly'' converge to thesteady state and that, simultaneously, the time-step sizes stabilize to alimiting size $dt_\infty$. In the companion to this article \cite{YPD_Part2},we linearize the SBDF2 scheme about the steady-state solution and demonstratethat the linearized scheme is conditionally stable. This conditional stabilityis the cause of the adaptive time-stepper's behaviour. While the adaptivetime-stepper based on the fully-implicit (BDF2) method is not subject to suchtime-step constraints, the required nonlinear solve yields run times that aresignificantly longer.

Adaptive Time-stepping Schemes for the Solution of the Poisson-Nernst-Planck Equations