A Meshfree Stochastic Algorithm for Solving Diffusion—Convection—Reaction Equations in Domains with Complex Geometry

Publication type Article
Status Published
Affiliation: Institute of Computational Mathematics and Mathematical Geophysics, Siberian Branch RAS
Address: Russian Federation
Journal nameDoklady Akademii nauk
EditionVolume 482 Issue 2

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 obtainned. 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.

Publication date06.11.2018
Cite   Download pdf To download PDF you should sign in
Размещенный ниже текст является ознакомительной версией и может не соответствовать печатной

views: 1435

Readers community rating: votes 0

1. L. Devroye, The series method for random variate generation and its application to the Kolmogorov-Smirnov distribution. American Journal of Mathematical and Management Sciences, 1(4) (1981), 359-379.

2. S. M. Ermakov, V. V. Nekrutkin and A. S. Sipin, Random Processes for Classical Equations of Mathematical Physics, Kluwer Academic Publishers, Dodrecht, 1989.

3. A. Friedman. Partial differential equations of parabolic type. Courier Dover Publications, 2008.

4. A. Haji-Sheikh and E. M. Sparrow, The floating random walk and its application to Monte Carlo solutions of heat equations, SIAM J. Appl. Math. v. 14 (1966), 2, 570–589.

5. K. Ito, G. Makkin. Diffuzionnye protsessy i ikh traektorii. Izd-vo Mir, Moskva, 1968.

6. P. Kloeden, E. Platen, H. Schurz. Numerical Solution of Stochastic Differential Equations. Springer, Heidelberg-Berlin, 2012.

7. M. E. Muller, Some continuous Monte Carlo methods for the Dirichlet problem, Ann. Math. Statist. 27 (1956), no. 3, 569–589.

8. K. K. Sabelfeld, Monte Carlo Methods in Boundary Value Problems, Springer, Berlin, 1991.

9. K. K. Sabelfeld and N. A. Simonov, Stochastic Methods for Boundary Value Problems. Numerics for High-imensional PDEs and Applications, De Gruyter, Berlin, 2016.

10. K. K. Sabelfeld, A mesh free floating random walk method for solving diffusion imaging problems, Statist. Probab. Lett. 121 (2017), 6–11.

11. K. K. Sabelfeld. Random walk on spheres method for solving drift-diffusion problems. Monte Carlo Methods Appl. 2016; 22 (4): 265-281.

Система Orphus