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Walk-on-Spheres Algorithm for Solving Boundary-Value Problems with Continuity Flux Conditions

In: Monte Carlo and Quasi-Monte Carlo Methods 2006

Author

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  • Nikolai Simonov

    (Institute of Computational Mathematics and Mathematical Geophysics SB RAS
    Florida State University, Institute of Molecular Biophysics)

Abstract

Summary We consider boundary-value problem for elliptic equations with constant coefficients related through the continuity conditions on the boundary between the domains. To take into account conditions involving the solution’s normal derivative, we apply a new mean-value relation written down at a boundary point. This integral relation is exact and provides a possibility to get rid of the bias caused by usually used finite-difference approximation. Randomization of this mean-value relation makes it possible to continue simulating walk-on-spheres trajectory after it hits the boundary. We prove the convergence of the algorithm and determine its rate. In conclusion, we present the results of some model computations.

Suggested Citation

  • Nikolai Simonov, 2008. "Walk-on-Spheres Algorithm for Solving Boundary-Value Problems with Continuity Flux Conditions," Springer Books, in: Alexander Keller & Stefan Heinrich & Harald Niederreiter (ed.), Monte Carlo and Quasi-Monte Carlo Methods 2006, pages 633-643, Springer.
    • Handle: RePEc:spr:sprchp:978-3-540-74496-2_38
    DOI: 10.1007/978-3-540-74496-2_38
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