A finite volume scheme with preconditioned Lanczos method for two-dimensional space-fractional reaction–diffusion equations
Yang, Qianqian, Turner, Ian, Moroney, Timothy J., & Liu, Fawang (2014) A finite volume scheme with preconditioned Lanczos method for two-dimensional space-fractional reaction–diffusion equations. Applied Mathematical Modelling, 38(15-16), pp. 3755-3762.
Fractional differential equations have been increasingly used as a powerful tool to model the non-locality and spatial heterogeneity inherent in many real-world problems. However, a constant challenge faced by researchers in this area is the high computational expense of obtaining numerical solutions of these fractional models, owing to the non-local nature of fractional derivatives. In this paper, we introduce a finite volume scheme with preconditioned Lanczos method as an attractive and high-efficiency approach for solving two-dimensional space-fractional reaction–diffusion equations. The computational heart of this approach is the efficient computation of a matrix-function-vector product f(A)bf(A)b, where A A is the matrix representation of the Laplacian obtained from the finite volume method and is non-symmetric. A key aspect of our proposed approach is that the popular Lanczos method for symmetric matrices is applied to this non-symmetric problem, after a suitable transformation. Furthermore, the convergence of the Lanczos method is greatly improved by incorporating a preconditioner. Our approach is show-cased by solving the fractional Fisher equation including a validation of the solution and an analysis of the behaviour of the model.
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|Item Type:||Journal Article|
|Keywords:||Fractional Laplacian, Matrix function, Krylov subspace, Preconditioner, Matrix transfer technique|
|Subjects:||Australian and New Zealand Standard Research Classification > MATHEMATICAL SCIENCES (010000) > NUMERICAL AND COMPUTATIONAL MATHEMATICS (010300) > Numerical Solution of Differential and Integral Equations (010302)|
|Divisions:||Past > QUT Faculties & Divisions > Faculty of Science and Technology
Current > Schools > School of Mathematical Sciences
|Copyright Owner:||Copyright 2014 Elsevier|
|Copyright Statement:||This is the author’s version of a work that was accepted for publication in Applied Mathematical Modelling. Changes resulting from the publishing process, such as peer review, editing, corrections, structural formatting, and other quality control mechanisms may not be reflected in this document. Changes may have been made to this work since it was submitted for publication. A definitive version was subsequently published in Applied Mathematical Modelling, [in press] DOI: 10.1016/j.apm.2014.02.005|
|Deposited On:||22 Jun 2014 22:36|
|Last Modified:||10 Aug 2016 20:32|
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