Random walk on spheres algorithm for solving transient drift-diffusion-reaction problems

2017 ◽  
Vol 23 (3) ◽  
Author(s):  
Karl K. Sabelfeld
Keyword(s):  
Random Walk ◽  
Diffusion Reaction ◽  
Drift Diffusion ◽  
Walk On Spheres Algorithm

AbstractWe suggest in this paper a Random Walk on Spheres (RWS) method for solving transient drift-diffusion-reaction problems which is an extension of our algorithm we developed recently [

A global random walk on spheres algorithm for transient heat equation and some extensions

2019 ◽  
Vol 25 (1) ◽  
pp. 85-96 ◽  
Author(s):  
Karl K. Sabelfeld
Keyword(s):  
Random Walk ◽  
Heat Equation ◽  
Symmetry Property ◽  
Forthcoming Paper ◽  
Diffusion Reaction ◽  
Advection Diffusion ◽  
The Green Function ◽  
Walk On Spheres Algorithm ◽  
Reaction Equations ◽  
Advection Diffusion Reaction Equations

AbstractWe suggest in this paper a global Random Walk on Spheres (gRWS) method for solving transient boundary value problems, which, in contrast to the classical RWS method, calculates the solution in any desired family ofmprescribed points. The method uses onlyNtrajectories in contrast tomNtrajectories in the conventional RWS algorithm. The idea is based on the symmetry property of the Green function and a double randomization approach. We present the gRWS method for the heat equation with arbitrary initial and boundary conditions, and the Laplace equation. Detailed description is given for 3D problems; the 2D problems can be treated analogously. Further extensions to advection-diffusion-reaction equations will be presented in a forthcoming paper.

Random walk on rectangles and parallelepipeds algorithm for solving transient anisotropic drift-diffusion-reaction problems

2019 ◽  
Vol 25 (2) ◽  
pp. 131-146 ◽  
Author(s):  
Karl K. Sabelfeld
Keyword(s):  
Random Walk ◽  
First Passage Time ◽  
Three Dimensional ◽  
Passage Time ◽  
Diffusion Reaction ◽  
Exit Point ◽  
Drift Diffusion ◽  
Continuity Equations ◽  
Reaction Equations ◽  
Reaction Problem

Abstract In this paper a random walk on arbitrary rectangles (2D) and parallelepipeds (3D) algorithm is developed for solving transient anisotropic drift-diffusion-reaction equations. The method is meshless, both in space and time. The approach is based on a rigorous representation of the first passage time and exit point distributions for arbitrary rectangles and parallelepipeds. The probabilistic representation is then transformed to a form convenient for stochastic simulation. The method can be used to calculate fluxes to any desired part of the boundary, from arbitrary sources. A global version of the method we call here as a stochastic expansion from cell to cell (SECC) algorithm for calculating the whole solution field is suggested. Application of this method to solve a system of transport equations for electrons and holes in a semicoductor is discussed. This system consists of the continuity equations for particle densities and a Poisson equation for electrostatic potential. To validate the method we have derived a series of exact solutions of the drift-diffusion-reaction problem in a three-dimensional layer presented in the last section in details.

Random walk on spheres method for solving anisotropic drift-diffusion problems

2018 ◽  
Vol 24 (1) ◽  
pp. 43-54 ◽  
Author(s):  
Irina Shalimova ◽  
Karl K. Sabelfeld
Keyword(s):  
Random Walk ◽  
Anisotropic Diffusion ◽  
Stochastic Simulation Algorithm ◽  
Local Variation ◽  
Diffusion Reaction ◽  
Drift Diffusion ◽  
Mesh Free ◽  
Electron Beam Induced Current ◽  
Defects In Semiconductors ◽  
Diffusion Problems

Abstract We suggest a random walk on spheres based stochastic simulation algorithm for solving drift-diffusion-reaction problems with anisotropic diffusion. The diffusion coefficients and the velocity vector vary in space, and the size of the walking spheres is adapted to the local variation of these functions. The method is mesh free and extremely efficient for calculation of fluxes to boundaries and the concentration of the absorbed particles inside the domain. Applications to cathodoluminescence (CL) and electron beam induced current (EBIC) methods for the analysis of dislocations and other defects in semiconductors are discussed.

Random walk on spheres method for solving anisotropic drift-diffusion problems

2018 ◽  
Vol 0 (0) ◽  
Author(s):  
Irina Shalimova ◽  
Karl K. Sabelfeld
Keyword(s):  
Random Walk ◽  
Anisotropic Diffusion ◽  
Stochastic Simulation Algorithm ◽  
Local Variation ◽  
Diffusion Reaction ◽  
Drift Diffusion ◽  
Mesh Free ◽  
Electron Beam Induced Current ◽  
Defects In Semiconductors ◽  
Diffusion Problems

AbstractWe suggest a random walk on spheres based stochastic simulation algorithm for solving drift-diffusion-reaction problems with anisotropic diffusion. The diffusion coefficients and the velocity vector vary in space, and the size of the walking spheres is adapted to the local variation of these functions. The method is mesh free and extremely efficient for calculation of fluxes to boundaries and the concentration of the absorbed particles inside the domain. Applications to cathodoluminescence (CL) and electron beam induced current (EBIC) methods for the analysis of dislocations and other defects in semiconductors are discussed.

Diffusion-reaction in one dimension

1988 ◽  
Vol 25 (04) ◽  
pp. 733-743 ◽  
Author(s):  
David Balding
Keyword(s):  
Brownian Motion ◽  
Random Walk ◽  
Particle Systems ◽  
Diffusion Reaction ◽  
One Dimension ◽  
One Dimensional ◽  
Diffusion Limited ◽  
Special Cases ◽  
Initial Arrangement ◽  
Number Of Particles

One-dimensional, periodic and annihilating systems of Brownian motions and random walks are defined and interpreted in terms of sizeless particles which vanish on contact. The generating function and moments of the number pairs of particles which have vanished, given an arbitrary initial arrangement, are derived in terms of known two-particle survival probabilities. Three important special cases are considered: Brownian motion with the particles initially (i) uniformly distributed and (ii) equally spaced on a circle and (iii) random walk on a lattice with initially each site occupied. Results are also given for the infinite annihilating particle systems obtained in the limit as the number of particles and the size of the circle or lattice increase. Application of the results to the theory of diffusion-limited reactions is discussed.

Diffusion-reaction in one dimension

1988 ◽  
Vol 25 (4) ◽  
pp. 733-743 ◽  
Author(s):  
David Balding
Keyword(s):  
Brownian Motion ◽  
Random Walk ◽  
Particle Systems ◽  
Diffusion Reaction ◽  
One Dimension ◽  
One Dimensional ◽  
Diffusion Limited ◽  
Special Cases ◽  
Initial Arrangement ◽  
Number Of Particles

One-dimensional, periodic and annihilating systems of Brownian motions and random walks are defined and interpreted in terms of sizeless particles which vanish on contact. The generating function and moments of the number pairs of particles which have vanished, given an arbitrary initial arrangement, are derived in terms of known two-particle survival probabilities. Three important special cases are considered: Brownian motion with the particles initially (i) uniformly distributed and (ii) equally spaced on a circle and (iii) random walk on a lattice with initially each site occupied. Results are also given for the infinite annihilating particle systems obtained in the limit as the number of particles and the size of the circle or lattice increase. Application of the results to the theory of diffusion-limited reactions is discussed.

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