From stochastic processes to numerical methods: a new scheme for solving reaction subdiffusion fractional partial differential equations

Article

Angstmann, C. N., Donnelly, I. C., Henry, B. I., Jacobs, B. A., Langlands, T. A. M. and Nichols, J. A.. 2016. "From stochastic processes to numerical methods: a new scheme for solving reaction subdiffusion fractional partial differential equations." Journal of Computational Physics. 307, pp. 508-534. https://doi.org/10.1016/j.jcp.2015.11.053
Article Title

From stochastic processes to numerical methods: a new scheme for solving reaction subdiffusion fractional partial differential equations

ERA Journal ID35103
Article CategoryArticle
AuthorsAngstmann, C. N. (Author), Donnelly, I. C. (Author), Henry, B. I. (Author), Jacobs, B. A. (Author), Langlands, T. A. M. (Author) and Nichols, J. A. (Author)
Journal TitleJournal of Computational Physics
Journal Citation307, pp. 508-534
Number of Pages27
Year2016
Place of PublicationNetherlands
ISSN0021-9991
1090-2716
Digital Object Identifier (DOI)https://doi.org/10.1016/j.jcp.2015.11.053
Web Address (URL)http://www.sciencedirect.com/science/article/pii/S0021999115007937
Abstract

We have introduced a new explicit numerical method, based on a discrete stochastic process, for solving a class of fractional partial differential equations that model reaction subdiffusion. The scheme is derived from the master equations for the evolution of the probability density of a sum of discrete time random walks. We show that the diffusion limit of the master equations recovers the fractional partial differential equation of interest. This limiting procedure guarantees the consistency of the numerical scheme. The positivity of the solution and stability results are simply obtained, provided that the underlying process is well posed. We also show that the method can be applied to standard reaction–diffusion equations. This work highlights the broader applicability of using discrete stochastic processes to provide numerical schemes for partial differential equations, including fractional partial differential equations.

Keywordsfractional diffusion; fractional reaction diffusion; anomalous diffusion; continuous time random walk; discrete time random walk; finite difference method
Contains Sensitive ContentDoes not contain sensitive content
ANZSRC Field of Research 2020490303. Numerical solution of differential and integral equations
490510. Stochastic analysis and modelling
Public Notes

File reproduced in accordance with the copyright policy of the publisher/author.

Byline AffiliationsUniversity of New South Wales
University of the Witwatersrand, South Africa
Computational Engineering and Science Research Centre
Institution of OriginUniversity of Southern Queensland
Funding source
Australian Research Council (ARC)
Grant ID
DP140101193
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