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 Faradaic reactions, and have applications in a number of fields. In this article, we develop an adaptive time-stepping scheme for the solution PNP-FBV system based on two methods: fully-implicit (VSBDF2) method, semi-implicit (VSSBDF2) method. We present simulations under both current voltage demonstrate ability to simulate large range parameters, including any value singular perturbation parameter ϵ. Many electrochemical systems interest are subject sudden changes forcing separated by periods constant or no forcing. time-stepper easily addresses such time-scale changes. When underlying dynamics is one that would solutions converge steady-state solution, observe VSSBDF2 method produces “nearly” that, simultaneously, time-step sizes stabilize limiting size dt∞. While not stability restrictions, required nonlinear solve incurs additional computational cost. profile methods identify regimes ϵ where favourable over other. Matlab code used work can be found at https://github.com/daveboat/vssimex_pnp.