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

Nikolai Simonov ()
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Nikolai Simonov: Institute of Computational Mathematics and Mathematical Geophysics SB RAS

A chapter in Monte Carlo and Quasi-Monte Carlo Methods 2006, 2008, pp 633-643 from Springer

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.

Keywords: Markov Chain; Random Walk; Neumann Problem; Normal Derivative; Integral Relation (search for similar items in EconPapers)
Date: 2008
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Persistent link: https://EconPapers.repec.org/RePEc:spr:sprchp:978-3-540-74496-2_38

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DOI: 10.1007/978-3-540-74496-2_38

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