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Random walk approximation of fractional-order multiscaling anomalous diffusion
Yong Zhang1, David A Benson, Mark M Meerschaert
1Department of Geology and Geological Engineering, Colorado School of Mines, Golden, CO 80401, USA.
Physical Review. E, Statistical, Nonlinear, and Soft Matter Physics
|October 10, 2006
Summary
Random walks approximate anomalous diffusion equations by considering scaling indexes and mixing measures. This particle-tracking method offers a flexible and efficient solution for complex, nonhomogeneous diffusion problems.
Area of Science:
- Physics
- Applied Mathematics
- Computational Science
Background:
- Anomalous diffusion phenomena are prevalent in various scientific fields.
- Existing numerical methods struggle with multiscaling and fractional-order diffusion equations.
- Understanding diffusion processes requires accurate modeling of diffusion coefficients and drift in heterogeneous media.
Purpose of the Study:
- To develop and validate a random walk method for approximating solutions to multiscaling, fractional-order, anomalous diffusion equations.
- To extend the particle-tracking algorithm for anomalous diffusion with spatially varying diffusion properties.
- To demonstrate the applicability of the random walk method for nonhomogeneous diffusion scenarios.
Main Methods:
- Development of a random walk framework incorporating matrix-order scaling indexes and a mixing measure.
- Extension of a particle-tracking algorithm using a streamline-projection technique for streamline-dependent mixing.
- Application of the random walk method to solve two forms of the multiscaling fractional diffusion equation.
- Numerical simulations of five distinct examples to showcase method performance.
Main Results:
- The random walk method effectively approximates solutions for multiscaling, fractional-order, anomalous diffusion.
- The particle-tracking algorithm successfully handles anomalous diffusion with streamline-dependent mixing measures.
- The developed method demonstrates flexibility, simplicity, and efficiency across various diffusion scenarios.
- The random walk approach is identified as a key method for solving nonhomogeneous diffusion equations.
Conclusions:
- Random walks provide a robust numerical tool for complex diffusion processes.
- The enhanced particle-tracking algorithm expands the scope of random walk applications in fluid dynamics and transport phenomena.
- This study highlights the significance of the random walk method in addressing challenging diffusion problems in science and engineering.
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