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 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...
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Main Authors | , , |
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Format | Journal Article |
Language | English |
Published |
29.03.2017
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Subjects | |
Online Access | Get full text |
DOI | 10.48550/arxiv.1703.10297 |
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Summary: | 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 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. |
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DOI: | 10.48550/arxiv.1703.10297 |