Random walk methods for Monte Carlo simulations of Brownian diffusion on a sphere
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Date
2019-02
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Abstract
This paper is focused on efficient Monte Carlo simulations of Brownian diffusion
effects in particle-based numerical methods for solving transport equations
on a sphere (or a circle). Using the heat equation as a model problem,
random walks are designed to emulate the action of the Laplace-Beltrami
operator without evolving or reconstructing the probability density function.
The intensity of perturbations is fitted to the value of the rotary diffusion
coefficient in the deterministic model. Simplified forms of Brownian motion
generators are derived for rotated reference frames, and several practical
approaches to generating random walks on a sphere are discussed. The alternatives
considered in this work include projections of Cartesian random
walks, as well as polar random walks on the tangential plane. In addition,
we explore the possibility of using look-up tables for the exact cumulative
probability of perturbations. Numerical studies are performed to assess the
practical utility of the methods under investigation.
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Keywords
Brownian diffusion on a sphere, Laplace-Beltrami operator, orientation probability density, Lagrangian modeling, random walk