A Study of the Numerical Stability of an ImEx Scheme with Application to the Poisson-Nernst-Planck Equations

TL;DR

This paper investigates the numerical stability of an implicit-explicit (ImEx) scheme applied to the Poisson-Nernst-Planck equations with boundary conditions relevant to ion transport, revealing stability characteristics and adaptive time-stepping behavior.

Contribution

It provides a linear stability analysis of the VSSBDF2 scheme for PNP-FBV equations, linking adaptive time-step behavior to conditional stability and analyzing the scheme's stability domain.

Findings

Adaptive time-stepper solutions nearly converge to steady state.

Time-step sizes stabilize at a limiting value due to conditional stability.

Stability domain exhibits corners and jump discontinuities.

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 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.