A direct simulation approach for the Poisson-Boltzmann equation using the Random Batch Method

The Poisson-Boltzmann equation is a nonlinear elliptic equation that describes how electronic potential changes in the screening layer when a cell is immersed in an ionic solution. It is the equilibrium state formed by many charged particles interacting with each other through Coulomb forces. We propose a direct simulation approach for the dynamics of the charged particles in the solution governed by the Poisson-Nernst-Planck equation and the Poisson-Boltzmann equation can be solved automatically when the particles reach the equilibrium. The interacting $N$ particle system is simulated in the truncated external domain. Directly simulating the $N$ particle system has an $\mathcal{O}(N^2)$ computational cost in each time step. By making use of the Random Batch Method (RBM) proposed by Jin et al [J. Comput. Phys., 400(1): 108877, 2020], the cost per time step is reduced remarkably to $\mathcal{O}(N)$. Another benefit of RBM is that the hard sphere potential is no longer needed in the simulation. An error analysis is provided for the truncation and numerical experiments are performed to validate the method. This particle method is preferable in two aspects: it is simple and effective in high dimensions, especially in non-symmetric cases or with complicated membrane or molecular surface. Meanwhile, direct simulation may be preferred as people may be interested in the dynamics of the physical process.