TY - JOUR

T1 - A Global Random Walk on Spheres algorithm for calculating the solution and its derivatives of the drift-diffusion-reaction equations

AU - Sabelfeld, Karl

AU - Kireeva, Anastasya

N1 - Funding Information:
This study was supported by the Russian Science Foundation under Grant 19‐11‐00019, in the part of the GRWS theory development, and the Mathematical Center in Akademgorodok, the agreement with the Ministry of Science and High Education of the Russian Federation Number 075‐15‐2019‐1675, in the part of simulations.
Publisher Copyright:
© 2021 John Wiley & Sons, Ltd.

PY - 2021/10/7

Y1 - 2021/10/7

N2 - A Global Random Walk on Spheres (GRWS) algorithm for calculating the solution and its derivatives of drift-diffusion-reaction equations in any desired set of points is suggested. The GRWS algorithm is able to find the solution and derivative fields in any family of points simultaneously, using only one ensemble of random walks which drastically decreases the computer time compared to the standard random walk-based methods. The method in its nature is stochastic and meshless; the cost is of the order (Formula presented.) independent of the space dimension and complexity of the boundary shape, where ε is the desired accuracy. The variance, accuracy, and the cost of the method suggested are given. For illustration, we present simulation results for exciton drift-diffusion-recombination transport in a 3D prism and compare them with the exact solution. Calculations are also given for derivatives of a 2D diffusion problem which show the same accuracy as for the solution itself; the simulations are compared against the exact results.

AB - A Global Random Walk on Spheres (GRWS) algorithm for calculating the solution and its derivatives of drift-diffusion-reaction equations in any desired set of points is suggested. The GRWS algorithm is able to find the solution and derivative fields in any family of points simultaneously, using only one ensemble of random walks which drastically decreases the computer time compared to the standard random walk-based methods. The method in its nature is stochastic and meshless; the cost is of the order (Formula presented.) independent of the space dimension and complexity of the boundary shape, where ε is the desired accuracy. The variance, accuracy, and the cost of the method suggested are given. For illustration, we present simulation results for exciton drift-diffusion-recombination transport in a 3D prism and compare them with the exact solution. Calculations are also given for derivatives of a 2D diffusion problem which show the same accuracy as for the solution itself; the simulations are compared against the exact results.

KW - adjoint Green function

KW - drift-diffusion-Poisson equation

KW - fundamental solution

KW - Global Random Walk algorithm

KW - Random Walk on Spheres

UR - http://www.scopus.com/inward/record.url?scp=85116466614&partnerID=8YFLogxK

U2 - 10.1002/mma.7861

DO - 10.1002/mma.7861

M3 - Article

AN - SCOPUS:85116466614

JO - Mathematical Methods in the Applied Sciences

JF - Mathematical Methods in the Applied Sciences

SN - 0170-4214

ER -