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Volume 27, Issue 5
Unconditional Positivity-Preserving and Energy Stable Schemes for a Reduced Poisson-Nernst-Planck System

Hailiang Liu & Wumaier Maimaitiyiming

DOI: 10.4208/cicp.OA-2019-0063

Commun. Comput. Phys., 27 (2020), pp. 1505-1529.

Published online: 2020-03

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  • Abstract

The Poisson-Nernst-Planck (PNP) system is a widely accepted model for simulation of ionic channels. In this paper, we design, analyze, and numerically validate a second order unconditional positivity-preserving scheme for solving a reduced PNP system, which can well approximate the three dimensional ion channel problem. Positivity of numerical solutions is proven to hold true independent of the size of time steps and the choice of the Poisson solver. The scheme is easy to implement without resorting to any iteration method. Several numerical examples further confirm the positivity-preserving property, and demonstrate the accuracy, efficiency, and robustness of the proposed scheme, as well as the fast approach to steady states.

  • Keywords

Biological channels, diffusion models, ion transport, positivity.

  • AMS Subject Headings

65N08, 65N12, 92C35

  • Copyright

COPYRIGHT: © Global Science Press

  • Email address

hliu@iastate.edu (Hailiang Liu)

wumaierm@iastate.edu (Wumaier Maimaitiyiming)

  • BibTex
  • RIS
  • TXT
@Article{CiCP-27-1505, author = {Liu , Hailiang and Maimaitiyiming , Wumaier}, title = {Unconditional Positivity-Preserving and Energy Stable Schemes for a Reduced Poisson-Nernst-Planck System}, journal = {Communications in Computational Physics}, year = {2020}, volume = {27}, number = {5}, pages = {1505--1529}, abstract = {

The Poisson-Nernst-Planck (PNP) system is a widely accepted model for simulation of ionic channels. In this paper, we design, analyze, and numerically validate a second order unconditional positivity-preserving scheme for solving a reduced PNP system, which can well approximate the three dimensional ion channel problem. Positivity of numerical solutions is proven to hold true independent of the size of time steps and the choice of the Poisson solver. The scheme is easy to implement without resorting to any iteration method. Several numerical examples further confirm the positivity-preserving property, and demonstrate the accuracy, efficiency, and robustness of the proposed scheme, as well as the fast approach to steady states.

}, issn = {1991-7120}, doi = {https://doi.org/10.4208/cicp.OA-2019-0063}, url = {http://global-sci.org/intro/article_detail/cicp/15767.html} }
TY - JOUR T1 - Unconditional Positivity-Preserving and Energy Stable Schemes for a Reduced Poisson-Nernst-Planck System AU - Liu , Hailiang AU - Maimaitiyiming , Wumaier JO - Communications in Computational Physics VL - 5 SP - 1505 EP - 1529 PY - 2020 DA - 2020/03 SN - 27 DO - http://doi.org/10.4208/cicp.OA-2019-0063 UR - https://global-sci.org/intro/article_detail/cicp/15767.html KW - Biological channels, diffusion models, ion transport, positivity. AB -

The Poisson-Nernst-Planck (PNP) system is a widely accepted model for simulation of ionic channels. In this paper, we design, analyze, and numerically validate a second order unconditional positivity-preserving scheme for solving a reduced PNP system, which can well approximate the three dimensional ion channel problem. Positivity of numerical solutions is proven to hold true independent of the size of time steps and the choice of the Poisson solver. The scheme is easy to implement without resorting to any iteration method. Several numerical examples further confirm the positivity-preserving property, and demonstrate the accuracy, efficiency, and robustness of the proposed scheme, as well as the fast approach to steady states.

Liu , Hailiang and Maimaitiyiming , Wumaier. (2020). Unconditional Positivity-Preserving and Energy Stable Schemes for a Reduced Poisson-Nernst-Planck System. Communications in Computational Physics. 27 (5). 1505-1529. doi:10.4208/cicp.OA-2019-0063
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