  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. Using an adaptive
time-stepper based on a second-order variable step-size, semi-implicit,
backward differentiation formula (VSSBDF2), we observe that when the underlying
dynamics is one that would have the solutions converge to a steady state
solution, the adaptive time-stepper produces solutions that "nearly" converge
to the steady state and that, simultaneously, the time-step sizes stabilize at
a limiting size $dt_\infty$. Linearizing the SBDF2 scheme about the steady
state solution, we demonstrate that the linearized scheme is conditionally
stable and that this is the cause of the adaptive time-stepper's behaviour.
Mesh-refinement, as well as a study of the eigenvectors corresponding to the
critical eigenvalues, demonstrate that the conditional stability is not due to
a time-step constraint caused by high-frequency contributions. We study the
stability domain of the linearized scheme and find that it can have corners as
well as jump discontinuities.
