| 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. |
