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A global random walk on spheres algorithm for transient heat equation and some extensions

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  • Sabelfeld Karl K.

    (Institute of Computational Mathematics and Mathematical Geophysics, Russian Acad. Sci, 630090, Lavrentiev Str. 6, Novosibirsk; and Novosibirsk State University, Russia)

Abstract

We suggest in this paper a global Random Walk on Spheres (gRWS) method for solving transient boundary value problems, which, in contrast to the classical RWS method, calculates the solution in any desired family of m prescribed points. The method uses only N trajectories in contrast to mN trajectories in the conventional RWS algorithm. The idea is based on the symmetry property of the Green function and a double randomization approach. We present the gRWS method for the heat equation with arbitrary initial and boundary conditions, and the Laplace equation. Detailed description is given for 3D problems; the 2D problems can be treated analogously. Further extensions to advection-diffusion-reaction equations will be presented in a forthcoming paper.

Suggested Citation

  • Sabelfeld Karl K., 2019. "A global random walk on spheres algorithm for transient heat equation and some extensions," Monte Carlo Methods and Applications, De Gruyter, vol. 25(1), pages 85-96, March.
    • Handle: RePEc:bpj:mcmeap:v:25:y:2019:i:1:p:85-96:n:5
    DOI: 10.1515/mcma-2019-2032
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    Cited by:

    1. Sabelfeld, Karl K. & Kireeva, Anastasya, 2020. "Stochastic simulation algorithms for solving a nonlinear system of drift–diffusion-Poisson equations of semiconductors," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 556(C).

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