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Random walk method for the two- and three-dimensional Laplace, Poisson and Helmholtz’s equations. (English) Zbl 0988.65004
Summary: 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 field solution. The method is based on the properties of diffusion processes, the Itô formula, the Dynkin formula, the Feynman-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 the method.

MSC:
65C50Other computational problems in probability
65C05Monte Carlo methods
35J05Laplacian operator, reduced wave equation (Helmholtz equation), Poisson equation
60G50Sums of independent random variables; random walks
60J60Diffusion processes
60J65Brownian motion
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