Random Walks for Solving Boundary-Value Problems with Flux Conditions

  • Nikolai A. Simonov
Part of the Lecture Notes in Computer Science book series (LNCS, volume 4310)


We consider boundary-value problems for elliptic equations with constant coefficients and apply Monte Carlo methods to solving these equations. To take into account boundary conditions involving solution’s normal derivative, we apply the new mean-value relation written down at boundary point. This integral relation is exact and provides a possibility to get rid of the bias caused by usually used finite-difference approximation. We consider Neumann and mixed boundary-value problems, and also the problem with continuity boundary conditions, which involve fluxes. Randomization of the 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.


Markov Chain Random Walk Integral Operator Flux Condition Integral Relation 
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© Springer Berlin Heidelberg 2007

Authors and Affiliations

  • Nikolai A. Simonov
    • 1
    • 2
  1. 1.Institute of Molecular Biophysics, Florida State University, Tallahassee, FLUSA
  2. 2.Institute of Computational Mathematics and Mathematical Geophysics SB RAS, NovosibirskRussia

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