A study of the numerical stability of an ImEx scheme with application to the Poisson-Nernst-Planck equations
| Autor: | F.P. Dawson, David Yan, M. C. Pugh |
|---|---|
| Rok vydání: | 2021 |
| Předmět: |
Backward differentiation formula
Steady state (electronics) FOS: Physical sciences Boundary (topology) 010103 numerical & computational mathematics 01 natural sciences Stability (probability) 65M12 65Z05 78A57 symbols.namesake FOS: Mathematics Applied mathematics Nernst equation Mathematics - Numerical Analysis Boundary value problem 0101 mathematics Eigenvalues and eigenvectors Mathematics Numerical Analysis Applied Mathematics Numerical Analysis (math.NA) Computational Physics (physics.comp-ph) 010101 applied mathematics Computational Mathematics symbols Physics - Computational Physics Numerical stability |
| Zdroj: | Applied Numerical Mathematics. 163:239-253 |
| ISSN: | 0168-9274 |
| DOI: | 10.1016/j.apnum.2021.01.019 |
| Popis: | 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. The earlier version, arXiv:1905.01368v1, also contained: a linear stability analysis of the SBDF2 scheme, a study of the effect of Richardson Extrapolation on numerical stability, and a study of the stability domain of the logistic equation. This is the 2nd of a pair of articles: "A Variable Step Size Implicit-Explicit Scheme for the Solution of the Poisson-Nernst-Planck Equations", is on arXiV |
| Databáze: | OpenAIRE |
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