Walk-on-Hemispheres first-passage algorithm

Due to the isomorphism between an electrostatic problem and the corresponding Brownian diffusion one, the induced charge density on a conducting surface by a charge is isomorphic to the first-passage probability of the diffusion initiated at the location of the charge. Based on the isomorphism, many diffusion algorithms such as "Walk-on-Spheres" (WOS), "Walk-on-Planes" and so on have been developed. Among them, for fast diffusion simulations WOS algorithm is generally applied with an [Formula: see text]-layer, which is used for diffusion convergence on the boundary but induces another error from the [Formula: see text]-layer in addition to the intrinsic Monte Carlo error. However, for a finite flat boundary it is possible to terminate a diffusion process via "Walk-on-Hemispheres" (WOH) algorithm without the [Formula: see text]-layer. In this paper, we implement and demonstrate this algorithm for the induced charge density distribution on parallel infinite planes when a unit charge is between the plates. In addition, we apply it to the mutual capacitance of two circular parallel plates. In both simulations, WOH algorithm shows much better performance than the previous WOS algorithm.