TY - JOUR

T1 - Mesh-free stochastic algorithms for systems of drift–diffusion–reaction equations and anisotropic diffusion flux calculations

AU - Sabelfeld, Karl

PY - 2020/7

Y1 - 2020/7

N2 - We suggest in this paper two new random walk based stochastic algorithms for solving high-dimensional PDEs for domains with complicated geometrical structure. The first one, a Random Walk on Spheres (RWS) algorithm is developed for solving systems of coupled drift–diffusion–reaction equations where the random walk is living both on randomly sampled spheres and inside the relevant balls. The second method suggested solves transient anisotropic diffusion equations, where the random walk is carried out on random rectangular parallelepipeds inside the domain. The two methods are mesh-free both in space and time, and are well applied to solve high-dimensional problems with complicated domains. The algorithms are based on tracking the trajectories of the diffusing particles exactly in accordance with the probabilistic distributions derived from the explicit representation of the relevant Green functions for a sphere and a parallelepiped. They can be conveniently used not only for the solutions, but also for a direct calculation of fluxes to any part of the boundary without calculating the whole solution in the domain. Applications to exciton transport in semiconductors and related cathodoluminescence imaging of a set of randomly distributed threading dislocations are presented.

AB - We suggest in this paper two new random walk based stochastic algorithms for solving high-dimensional PDEs for domains with complicated geometrical structure. The first one, a Random Walk on Spheres (RWS) algorithm is developed for solving systems of coupled drift–diffusion–reaction equations where the random walk is living both on randomly sampled spheres and inside the relevant balls. The second method suggested solves transient anisotropic diffusion equations, where the random walk is carried out on random rectangular parallelepipeds inside the domain. The two methods are mesh-free both in space and time, and are well applied to solve high-dimensional problems with complicated domains. The algorithms are based on tracking the trajectories of the diffusing particles exactly in accordance with the probabilistic distributions derived from the explicit representation of the relevant Green functions for a sphere and a parallelepiped. They can be conveniently used not only for the solutions, but also for a direct calculation of fluxes to any part of the boundary without calculating the whole solution in the domain. Applications to exciton transport in semiconductors and related cathodoluminescence imaging of a set of randomly distributed threading dislocations are presented.

KW - A system of drift–diffusion–reaction equations

KW - Anisotropic diffusion

KW - First passage algorithms

KW - Random walk on cubes

KW - Random walk on spheres and balls

UR - http://www.scopus.com/inward/record.url?scp=85082687282&partnerID=8YFLogxK

U2 - 10.1016/j.probengmech.2020.103065

DO - 10.1016/j.probengmech.2020.103065

M3 - Article

AN - SCOPUS:85082687282

VL - 61

JO - Probabilistic Engineering Mechanics

JF - Probabilistic Engineering Mechanics

SN - 0266-8920

M1 - 103065

ER -