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

TL;DR

This paper develops adaptive time-stepping schemes, including BDF2 and SBDF2 methods, for solving Poisson-Nernst-Planck equations with boundary conditions, demonstrating their effectiveness and stability in simulating ion transport dynamics.

Contribution

It introduces and compares adaptive BDF2 and SBDF2 schemes for PNP-FBV equations, analyzing their stability, efficiency, and ability to handle large parameter ranges.

Findings

SBDF2 nearly converges to steady state with stabilized time steps

BDF2 requires longer run times due to nonlinear solves

Adaptive schemes effectively simulate ion transport with various parameters

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 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.