Volume 11, Issue 5
A Reduced Basis Method for the Nonlinear Poisson-Boltzmann Equation

Lijie Ji, Yanlai Chen & Zhenli Xu

Adv. Appl. Math. Mech., 11 (2019), pp. 1200-1218.

Published online: 2019-06

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  • Abstract

In numerical simulations of many charged systems at the micro/nano scale, a common theme is the repeated resolution of the Poisson-Boltzmann equation. This task proves challenging, if not entirely infeasible, largely due to the nonlinearity of the equation and the high dimensionality of the physical and parametric domains with the latter emulating the system configuration. In this paper, we for the first time adapt a mathematically rigorous and computationally efficient model order reduction paradigm, the so-called reduced basis method (RBM), to mitigate this challenge. We adopt a finite difference method as the mandatory underlying scheme to produce the high-fidelity numerical solutions of the Poisson-Boltzmann equation upon which the fast RBM algorithm is built and its performance is measured against. Numerical tests presented in this paper demonstrate the high efficiency and accuracy of the fast algorithm, the reliability of its error estimation, as well as its capability in effectively capturing the boundary layer.

  • Keywords

Reduced order modeling, reduced basis method, Poisson-Boltzmann equation, differential capacitance.

  • AMS Subject Headings

65M10, 78A48

  • Copyright

COPYRIGHT: © Global Science Press

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@Article{AAMM-11-1200, author = {}, title = {A Reduced Basis Method for the Nonlinear Poisson-Boltzmann Equation}, journal = {Advances in Applied Mathematics and Mechanics}, year = {2019}, volume = {11}, number = {5}, pages = {1200--1218}, abstract = {

In numerical simulations of many charged systems at the micro/nano scale, a common theme is the repeated resolution of the Poisson-Boltzmann equation. This task proves challenging, if not entirely infeasible, largely due to the nonlinearity of the equation and the high dimensionality of the physical and parametric domains with the latter emulating the system configuration. In this paper, we for the first time adapt a mathematically rigorous and computationally efficient model order reduction paradigm, the so-called reduced basis method (RBM), to mitigate this challenge. We adopt a finite difference method as the mandatory underlying scheme to produce the high-fidelity numerical solutions of the Poisson-Boltzmann equation upon which the fast RBM algorithm is built and its performance is measured against. Numerical tests presented in this paper demonstrate the high efficiency and accuracy of the fast algorithm, the reliability of its error estimation, as well as its capability in effectively capturing the boundary layer.

}, issn = {2075-1354}, doi = {https://doi.org/10.4208/aamm.OA-2018-0188}, url = {http://global-sci.org/intro/article_detail/aamm/13207.html} }
TY - JOUR T1 - A Reduced Basis Method for the Nonlinear Poisson-Boltzmann Equation JO - Advances in Applied Mathematics and Mechanics VL - 5 SP - 1200 EP - 1218 PY - 2019 DA - 2019/06 SN - 11 DO - http://dor.org/10.4208/aamm.OA-2018-0188 UR - https://global-sci.org/intro/article_detail/aamm/13207.html KW - Reduced order modeling, reduced basis method, Poisson-Boltzmann equation, differential capacitance. AB -

In numerical simulations of many charged systems at the micro/nano scale, a common theme is the repeated resolution of the Poisson-Boltzmann equation. This task proves challenging, if not entirely infeasible, largely due to the nonlinearity of the equation and the high dimensionality of the physical and parametric domains with the latter emulating the system configuration. In this paper, we for the first time adapt a mathematically rigorous and computationally efficient model order reduction paradigm, the so-called reduced basis method (RBM), to mitigate this challenge. We adopt a finite difference method as the mandatory underlying scheme to produce the high-fidelity numerical solutions of the Poisson-Boltzmann equation upon which the fast RBM algorithm is built and its performance is measured against. Numerical tests presented in this paper demonstrate the high efficiency and accuracy of the fast algorithm, the reliability of its error estimation, as well as its capability in effectively capturing the boundary layer.

Lijie Ji, Yanlai Chen & Zhenli Xu. (2019). A Reduced Basis Method for the Nonlinear Poisson-Boltzmann Equation. Advances in Applied Mathematics and Mechanics. 11 (5). 1200-1218. doi:10.4208/aamm.OA-2018-0188
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