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The simplest random walks for the Dirichlet problem. (English. Russian original) Zbl 1038.60066
Theory Probab. Appl. 47, No. 1, 53-68 (2002); translation from Teor. Veroyatn. Primen. 47, No. 1, 39-58 (2002).
The authors consider the Dirichlet problem for parabolic and elliptic partial differential equations (pdes) on a bounded domain \(G\) in \({\mathbb R}^d\). They assume the existence of a classical solution, strict ellipticity and sufficient smoothness of the coefficient functions. The authors propose Monte Carlo methods for the numerical approximation of the solution of the pde. That is, they suggest to simulate trajectories of an associate system of stochastic ordinary differential equations (sodes) and then calculate the solution of the pde via its probabilistic representation. The method chosen to approximate the system of sodes is the weak explicit Euler method with the simplest simulation of noise. An algorithm is proposed to deal with trajectories reaching the boundary of the domain. The authors give a convergence analysis of the proposed method and discuss some variations of the method. Numerical examples conclude the article.

MSC:
60H35 Computational methods for stochastic equations (aspects of stochastic analysis)
60H10 Stochastic ordinary differential equations (aspects of stochastic analysis)
60J60 Diffusion processes
65C05 Monte Carlo methods
65C30 Numerical solutions to stochastic differential and integral equations
65N99 Numerical methods for partial differential equations, boundary value problems
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