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

JOURNAL OF COMPUTATIONAL PHYSICS

卷:307

Article

Angstmann, C. N.¹  Donnelly, I. C.¹  Henry, B. I.¹  Jacobs, B. A.^2,3  Langlands, T. A. M.⁴  Nichols, J. A.¹ 

[1] UNSW Australia, Sch Math & Stat, Sydney, NSW 2052, Australia

[2] Univ Witwatersrand, Sch Comp Sci & Appl Math, Private Bag 3, ZA-2050 Johannesburg, South Africa

[3] DST NRF Ctr Excellence Math & Stat Sci CoE MaSS, Johannesburg, South Africa

[4] Univ So Queensland, Dept Math & Comp, Toowoomba, Qld 4350, Australia

关键词: Fractional diffusion;    Fractional reaction diffusion;    Anomalous diffusion;    Continuous time random walk;    Discrete time random walk;    Finite difference method;   

DOI  :  10.1016/j.jcp.2015.11.053

来源: Elsevier

【 摘 要 】

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. Crown Copyright (C) 2015 Published by Elsevier Inc. All rights reserved.

【 授权许可】

Free