A study of the numerical stability of an ImEx scheme with application to 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 wide variety of fields. We solve the PNP-FBV equations using an adaptive time-stepper based on a second-order variable step-size, semi-implicit, backward differentiation formula (VSSBDF2). When the underlying dynamics are such that the solutions converge to a steady-state solution, we observe that the adaptive time-stepper produces solutions that “nearly” converge to the steady state and that, simultaneously, the time-step sizes stabilize to a limiting size dt∞. 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 behavior. 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 restriction caused by high-frequency contributions. We study the stability domain of the linearized scheme and find that its boundary can have corners as well as jump discontinuities.