Positive and energy stable schemes for Poisson-Nernst-Planck equations and related models

dc.contributor.advisor Hailiang Liu
dc.contributor.author Maimaitiyiming, Wumaier
dc.contributor.department Mathematics
dc.date 2020-09-23T19:13:05.000
dc.date.accessioned 2021-02-25T21:35:15Z
dc.date.available 2021-02-25T21:35:15Z
dc.date.copyright Sat Aug 01 00:00:00 UTC 2020
dc.date.embargo 2020-09-10
dc.date.issued 2020-01-01
dc.description.abstract <p>In this thesis, we design, analyze, and numerically validate positive and energy-</p> <p>dissipating schemes for solving Poisson-Nernst-Planck (PNP) equations and Fokker-Planck (FP) equations with interaction potentials. These equations play an important role in modeling the dynamics of charged particles in semiconductors and biological ion channels, as well as in other applications. These model equations are nonlinear/nonlocal gradient flows in density space, and their explicit solutions are rarely available; however, solutions to such problems feature intrinsic properties such as (i) solution positivity, (ii) mass conservation, and (iii) energy dissipation. These physically relevant properties are highly desirable to be preserved at the discrete level with the least time-step restrictions.</p> <p>We first construct our schemes for a reduced PNP model, then extend to multi-dimensional PNP equations and a class of FP equations with interaction potentials. The common strategies in the construction of the baseline schemes include two ingredients: (i) reformulation of each underlying model so that the resulting system is more suitable for constructing positive schemes, and (ii) integration of semi-implicit time discretization and central spatial discretization. For each model equation, we show that the semi-discrete schemes (continuous in time) preserve all three solution properties (positivity, mass conservation, and energy dissipation). The fully discrete first order schemes preserve solution positivity and mass conservation for arbitrary time steps. Moreover, there exists a discrete energy function which dissipates along time marching with an $O(1)$ bound on time steps.</p> <p>We show that the second order (in both time and space) schemes preserve solution positivity for suitably small time steps; for larger time steps, we apply a local limiter to restore the solution positivity. We prove that such limiter preserves local mass and does not destroy the approximation accuracy. In addition, the limiter provides a reliable way of restoring solution positivity for other high order conservative finite difference or finite volume schemes.</p> <p>Both the first and second order schemes are linear and can be efficiently implemented without resorting to any iteration method. The second order schemes are only slight modifications of the first order schemes. Computational costs of a single time step for first and second order schemes are similar, hence our second-order in time schemes are efficient than the first-order in time schemes, given a larger time step could be utilized (to save computational cost). We conduct extensive numerical tests that support our theoretical results and illustrate the accuracy, efficiency, and capacity to preserve the solution properties of our schemes.</p>
dc.format.mimetype application/pdf
dc.identifier archive/lib.dr.iastate.edu/etd/18177/
dc.identifier.articleid 9184
dc.identifier.contextkey 19236746
dc.identifier.doi https://doi.org/10.31274/etd-20200902-96
dc.identifier.s3bucket isulib-bepress-aws-west
dc.identifier.submissionpath etd/18177
dc.identifier.uri https://dr.lib.iastate.edu/handle/20.500.12876/94329
dc.language.iso en
dc.source.bitstream archive/lib.dr.iastate.edu/etd/18177/Maimaitiyiming_iastate_0097E_18912.pdf|||Fri Jan 14 21:37:58 UTC 2022
dc.subject.keywords Fokker-Planck equations
dc.subject.keywords Poisson-Nernst-Planck equations
dc.title Positive and energy stable schemes for Poisson-Nernst-Planck equations and related models
dc.type article
dc.type.genre dissertation
dspace.entity.type Publication
relation.isOrgUnitOfPublication 82295b2b-0f85-4929-9659-075c93e82c48
thesis.degree.discipline Applied Mathematics
thesis.degree.level dissertation
thesis.degree.name Doctor of Philosophy
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