Random walk method for the two- and three-dimensional Laplace, Poisson and Helmholtzʹs equations

The random walk method (RWM) is developed here for solving the Laplace, Poisson, and Helmholtz equations in two and three dimensions. The RWM is a local method, i.e. the solution at an arbitrary point can be determined without having to obtain the complete Aeld solution. The method is based on the properties of diBusion processes, the Itˆo formula, theDynkin formula, theFe ynman–Kac functional, and Monte Carlo simulation. Simplicity, stability, accuracy, and generality are the main features of the proposed method. The RWK is inherently parallel and this fact has been fully exploited in this paper. Extensive numerical results have been presented in order to understand the various parameters involved in theme thod