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Random walk on spheres method for solving drift-diffusion problems

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  • Sabelfeld Karl K.

    (Institute of Computational Mathematics and Mathematical Geophysics, Russian Academy of Sciences, Novosibirsk, Russian Federation)

Abstract

The well-known random walk on spheres method (RWS) for the Laplace equation is here extended to drift-diffusion problems. First we derive a generalized spherical mean value relation which is an extension of the classical integral mean value relation for the Laplace equation. Next we give a probabilistic interpretation of the kernel. The distribution on the sphere generated by this kernel is then related to the von Mises–Fisher distribution on the sphere which can be efficiently simulated. The rigorous expressions are given for the case of constant velocity drift, but the algorithm is then extended to solve drift-diffusion problems with arbitrary varying drift velocity vector. Applications to cathodoluminescence and EBIC imaging of defects and dislocations in semiconductors are discussed.

Suggested Citation

  • Sabelfeld Karl K., 2016. "Random walk on spheres method for solving drift-diffusion problems," Monte Carlo Methods and Applications, De Gruyter, vol. 22(4), pages 265-275, December.
    • Handle: RePEc:bpj:mcmeap:v:22:y:2016:i:4:p:265-275:n:6
    DOI: 10.1515/mcma-2016-0118
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    References listed on IDEAS

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    1. Sabelfeld, Karl K., 2017. "A mesh free floating random walk method for solving diffusion imaging problems," Statistics & Probability Letters, Elsevier, vol. 121(C), pages 6-11.
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    Cited by:

    1. Sabelfeld, Karl K. & Kireeva, Anastasya, 2020. "Stochastic simulation algorithms for solving a nonlinear system of drift–diffusion-Poisson equations of semiconductors," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 556(C).

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    1. Sabelfeld, Karl K. & Kireeva, Anastasya, 2020. "Stochastic simulation algorithms for solving a nonlinear system of drift–diffusion-Poisson equations of semiconductors," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 556(C).
    2. Karl K. Sabelfeld & Anastasia E. Kireeva, 2022. "Stochastic Simulation Algorithms for Solving Transient Anisotropic Diffusion-recombination Equations and Application to Cathodoluminescence Imaging," Methodology and Computing in Applied Probability, Springer, vol. 24(4), pages 3029-3048, December.

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