Adaptive time-stepping schemes for the solution of the Poisson-Nernst-Planck equations

Abstract

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 system based on two time-stepping methods: a fully-implicit (VSBDF2) method, and a semi-implicit (VSSBDF2) 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 ϵ. Many electrochemical systems of interest are subject to sudden changes in forcing separated by periods with constant or no forcing. The adaptive time-stepper easily addresses such time-scale changes. 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 VSSBDF2 method produces solutions that “nearly” converge to the steady-state solution and that, simultaneously, the time-step sizes stabilize to a limiting size dt∞. While the adaptive time-stepper based on the fully-implicit (VSBDF2) method is not subject to such time-step stability restrictions, the required nonlinear solve incurs additional computational cost. We profile both methods to identify regimes of the perturbation parameter ϵ where one method is favourable over the other. The Matlab code used in this work can be found at https://github.com/daveboat/vssimex_pnp.