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Volume 32, Issue 1
Some Random Batch Particle Methods for the Poisson-Nernst-Planck and Poisson-Boltzmann Equations

Lei Li, Jian-Guo Liu & Yijia Tang

Commun. Comput. Phys., 32 (2022), pp. 41-82.

Published online: 2022-07

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  • Abstract

We consider in this paper random batch interacting particle methods for solving the Poisson-Nernst-Planck (PNP) equations, and thus the Poisson-Boltzmann (PB) equation as the equilibrium, in the external unbounded domain. To justify the simulation in a truncated domain, an error estimate of the truncation is proved in the symmetric cases for the PB equation. Then, the random batch interacting particle methods are introduced which are $\mathcal{O}(N)$ per time step. The particle methods can not only be considered as a numerical method for solving the PNP and PB equations, but also can be used as a direct simulation approach for the dynamics of the charged particles in solution. The particle methods are preferable due to their simplicity and adaptivity to complicated geometry, and may be interesting in describing the dynamics of the physical process. Moreover, it is feasible to incorporate more physical effects and interactions in the particle methods and to describe phenomena beyond the scope of the mean-field equations.

  • AMS Subject Headings

35Q92, 35Q84, 65N75

  • Copyright

COPYRIGHT: © Global Science Press

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@Article{CiCP-32-41, author = {Li , LeiLiu , Jian-Guo and Tang , Yijia}, title = {Some Random Batch Particle Methods for the Poisson-Nernst-Planck and Poisson-Boltzmann Equations}, journal = {Communications in Computational Physics}, year = {2022}, volume = {32}, number = {1}, pages = {41--82}, abstract = {

We consider in this paper random batch interacting particle methods for solving the Poisson-Nernst-Planck (PNP) equations, and thus the Poisson-Boltzmann (PB) equation as the equilibrium, in the external unbounded domain. To justify the simulation in a truncated domain, an error estimate of the truncation is proved in the symmetric cases for the PB equation. Then, the random batch interacting particle methods are introduced which are $\mathcal{O}(N)$ per time step. The particle methods can not only be considered as a numerical method for solving the PNP and PB equations, but also can be used as a direct simulation approach for the dynamics of the charged particles in solution. The particle methods are preferable due to their simplicity and adaptivity to complicated geometry, and may be interesting in describing the dynamics of the physical process. Moreover, it is feasible to incorporate more physical effects and interactions in the particle methods and to describe phenomena beyond the scope of the mean-field equations.

}, issn = {1991-7120}, doi = {https://doi.org/10.4208/cicp.OA-2021-0159}, url = {http://global-sci.org/intro/article_detail/cicp/20788.html} }
TY - JOUR T1 - Some Random Batch Particle Methods for the Poisson-Nernst-Planck and Poisson-Boltzmann Equations AU - Li , Lei AU - Liu , Jian-Guo AU - Tang , Yijia JO - Communications in Computational Physics VL - 1 SP - 41 EP - 82 PY - 2022 DA - 2022/07 SN - 32 DO - http://doi.org/10.4208/cicp.OA-2021-0159 UR - https://global-sci.org/intro/article_detail/cicp/20788.html KW - Interacting particle systems, Coulomb interaction, reflecting stochastic differential equation, charge reversal phenomenon, singular-regular decomposition. AB -

We consider in this paper random batch interacting particle methods for solving the Poisson-Nernst-Planck (PNP) equations, and thus the Poisson-Boltzmann (PB) equation as the equilibrium, in the external unbounded domain. To justify the simulation in a truncated domain, an error estimate of the truncation is proved in the symmetric cases for the PB equation. Then, the random batch interacting particle methods are introduced which are $\mathcal{O}(N)$ per time step. The particle methods can not only be considered as a numerical method for solving the PNP and PB equations, but also can be used as a direct simulation approach for the dynamics of the charged particles in solution. The particle methods are preferable due to their simplicity and adaptivity to complicated geometry, and may be interesting in describing the dynamics of the physical process. Moreover, it is feasible to incorporate more physical effects and interactions in the particle methods and to describe phenomena beyond the scope of the mean-field equations.

Lei Li, Jian-Guo Liu & Yijia Tang. (2022). Some Random Batch Particle Methods for the Poisson-Nernst-Planck and Poisson-Boltzmann Equations. Communications in Computational Physics. 32 (1). 41-82. doi:10.4208/cicp.OA-2021-0159
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