In a series of papers, the authors have been developing last‐passage algorithms such as last‐passage algorithm and off‐centered last‐passage algorithm on a flat conducting surface, quadrupole last‐passage algorithm on an L‐shaped conducting surface, and last‐passage algorithm on a spherical conducting surface, held at (non)constant potentials. All the previous last‐passage algorithms can compute charge density at a specific point on a flat or spherical conducting surface only. In this paper, the last‐passage algorithms on any smooth (convex) surface for charge density at a specific point are further developed where a tangent plane can be constructed and a sphere can be put. The algorithm for charge density on a sphere, on a circular plate, and on the unit cube held at unit potential is demonstrated. The results show excellent agreements with theoretical or other simulation ones.

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 $$\varepsilon $$ ε -layer, which is used for diffusion convergence on the boundary but induces another error from the $$\varepsilon $$ ε -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 $$\varepsilon $$ ε -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.