Random walk on semi-cylinders for diffusion problems with mixed Dirichlet–Robin boundary conditions

Abstract

Abstract We suggest random walk on semi-infinite cylinders methods for solving interior and exterior diffusion problems with different types of boundary conditions which include mixed Dirichlet, Neumann, and Robin boundary conditions on different parts of the boundary. Based on probabilistic interpretation of the diffusion process, stochastic simulation algorithms take into account specific features of each boundary condition to optimally adjust the Markov chain distribution on the relevant boundary parts. In contrast to the conventional direct trajectory tracking method, the new method avoids to simulate the diffusion trajectories. Instead, it exploits exact probabilities of different events like the first passage, splitting, and survival probabilities inside the semi-infinite cylinders, depending on the domain and its boundary structure. Applications to diffusion imaging methods like the cathodoluminescence (CL) and electron beam induced current (EBIC) semiconductor analysis techniques performed in scanning electron and transmission microscopes, are discussed.