TY - JOUR
T1 - Random walk methods for Monte Carlo simulations of Brownian diffusion on a sphere
AU - Novikov, A.
AU - Kuzmin, D.
AU - Ahmadi, O.
N1 - Publisher Copyright:
© 2019 Elsevier Inc.
PY - 2020/1/1
Y1 - 2020/1/1
N2 - 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.
AB - 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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U2 - 10.1016/j.amc.2019.124670
DO - 10.1016/j.amc.2019.124670
M3 - Article
AN - SCOPUS:85071247839
SN - 0096-3003
VL - 364
JO - Applied Mathematics and Computation
JF - Applied Mathematics and Computation
M1 - 124670
ER -