Doklady Mathematics

, Volume 98, Issue 2, pp 435–438 | Cite as

A Mesh Free Stochastic Algorithm for Solving Diffusion–Convection–Reaction Equations on Complicated Domains

  • K. K. SabelfeldEmail author


A mesh free stochastic algorithm for solving transient diffusion–convection–reaction problems on domains with complicated structure is suggested. For the solutions of this kind of equations exact representations of the survival probabilities, the probability densities of the first passage time and position on a sphere are obtained. Based on these representations we construct a stochastic algorithm which is simple in implementaion for solving one- and three-dimensional diffusion–convection–reaction equations. The method is continuous both in space and time, and its advantages are particularly well manifested in solving problems on complicated domains, calculating fluxes to parts of the boundary, and other integral functionals, for instance, the total concentration of the particles which have been reacted to a time instant t.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    L. Devroye, Am. J. Math. Manage. Sci. 1 (4) 359–379 (1981).Google Scholar
  2. 2.
    S. M. Ermakov, V. V. Nekrutkin and A. S. Sipin, Random Processes for Classical Equations of Mathematical Physics (Kluwer Academic, Dordrecht, 1989).CrossRefzbMATHGoogle Scholar
  3. 3.
    A. Friedman, Partial Differential Equations of Parabolic Type (Dover, Mineola, NY, 2008).Google Scholar
  4. 4.
    A. Haji-Sheikh and E. M. Sparrow, SIAM J. Appl. Math. 14 (2), 370–379 (1966).MathSciNetCrossRefGoogle Scholar
  5. 5.
    K. Ito and P. Mckean, Diffusion Processes and Their Sample Paths (Springer, Berlin, 1965).zbMATHGoogle Scholar
  6. 6.
    P. Kloeden, E. Platen, and H. Schurz, Numerical Solution of Stochastic Differential Equations (Springer, Heidelberg, 2012).zbMATHGoogle Scholar
  7. 7.
    M. E. Muller, Ann. Math. Stat. 27 (3), 569–589 (1956).CrossRefGoogle Scholar
  8. 8.
    K. K. Sabelfeld, Monte Carlo Methods in Boundary Value Problems (Springer, Berlin, 1991).Google Scholar
  9. 9.
    K. K. Sabelfeld and N. A. Simonov, Stochastic Methods for Boundary Value Problems: Numerics for High-Dimensional PDEs and Applications (De Gruyter, Berlin, 2016).CrossRefzbMATHGoogle Scholar
  10. 10.
    K. K. Sabelfeld, Stat. Probab. Lett. 121, 6–11 (2017).CrossRefGoogle Scholar
  11. 11.
    K. K. Sabelfeld, Monte Carlo Methods Appl. 22 (4), 265–281 (2016).MathSciNetCrossRefGoogle Scholar

Copyright information

© Pleiades Publishing, Ltd. 2018

Authors and Affiliations

  1. 1.Institute of Computational Mathematics and Mathematical GeophysicsRussian Academy of SciencesNovosibirskRussia

Personalised recommendations