435 ISSN 1064-5624, Doklady Mathematics, 2018, Vol. 98, No. 2, pp. 435–438. © Pleiades Publishing, Ltd., 2018. Original Russian Text © K.K. Sabelfeld, 2018, published in Doklady Akademii Nauk, 2018, Vol. 482, No. 2. A Mesh Free Stochastic Algorithm for Solving Diffusion–Convection–Reaction Equations on Complicated Domains 1 K. K. Sabelfeld Presented by Academician of the RAS A.N. Konovalov March 13, 2018 Received March 19, 2018 Abstract—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 represen- tations 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 imple- mentaion 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 . DOI: 10.1134/S1064562418060108 t Stochastic methods for solving boundary value problems for parabolic and elliptic differential equa- tions are based on probabilistic representations in the form of mathematical expectations in the space of dif- fusion processes [5]. Implementation of stochastic algorithms is then carried out by numerical solution of stochastic differential equations governing the trajec- tories of the diffusion processes [6]. However this approach needs constructions of temporal and spatial meshes which complicates the implementation on complicated domains and significantly reduces the efficiency of the method. For equations with constant coefficients a Random Walk on Spheres (RWS) algo- rithm is extremely efficient, it was first suggested in [7] for the Laplace equation, and in [4] for the heat equa- tion. This class of methods is mesh free, and is effi- ciently implemened on complicated domains. The RWS method was then extended on other equations mainly related to the Laplace equation and isotropic diffusion [2, 8, 9]. In all these studies the isotropy property of the Wiener process is used. In practice, when dealing with convection, one uses an approxi- mation by adding a small drift on a small time step. This however again brings the method back to the class 1 The article was translated by the author. of methods with meshes, and introduces an additional error related to an artificial separation of the diffusion and convection transfer. In this paper we suggest a mesh free RWS method for solving high-dimensional diffusion-convection-reaction problems which is based on a simple idea: we derive an exact distribution (in time and position) of the process with constant convection on an arbitrary sphere. This enables to carry out exact simulations of the particle transport in space and time governed by the diffusion–convec- tion–reaction equations. In the case when the diffu- sion coefficient, the convection and reaction terms are varying in space the random walking spheres are cho- sen small enough, and the RWS method remains mesh free. 1. DIFFUSION–CONVECTION–REACTION EQUATION Our goal is to construct a mesh free algorithm for solving a transient diffusion–convection–reaction equation based on simulation of particle trajectories governed by this equation. We call the random particle trajectory also as a diffusion–convection–reaction process. Physically, these processes describe the motion of a particle with a diffusion mobility charac- terized by the diffusion coefficient in a given velocity field , and under the reaction (say, chem- ical reactions) rate . We first consider for sim- plicity a one-dimensional case: we consider on a direct equation for the Green function () D x () vx λ 2 () x , [0 ] L MATHEMATICS Institute of Computational Mathematics and Mathematical Geophysics, Russian Academy of Sciences, Novosibirsk, 630090 Russia e-mail: karl@osmf.sscc.ru