Some random batch particle methods for the Poisson-Nernst-Planck and Poisson-Boltzmann equations

We consider in this paper random batch interacting particle methods for solving the Poisson-Nernst-Planck (PNP) equation, 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 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 its simplicity and adaptivity to complicated geometry of the surfaces, and may be interesting in describing the dynamics of the physical process.