Efficient solution of two-sided nonlinear space-fractional diffusion equations using fast Poisson preconditioners

Moroney, Timothy J. & Yang, Qianqian (2013) Efficient solution of two-sided nonlinear space-fractional diffusion equations using fast Poisson preconditioners. Journal of Computational Physics, 246, pp. 304-317.

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Abstract

We develop a fast Poisson preconditioner for the efficient numerical solution of a class of two-sided nonlinear space fractional diffusion equations in one and two dimensions using the method of lines. Using the shifted Gr¨unwald finite difference formulas to approximate the two-sided(i.e. the left and right Riemann-Liouville) fractional derivatives, the resulting semi-discrete nonlinear systems have dense Jacobian matrices owing to the non-local property of fractional derivatives. We employ a modern initial value problem solver utilising backward differentiation formulas and Jacobian-free Newton-Krylov methods to solve these systems. For efficient performance of the Jacobianfree Newton-Krylov method it is essential to apply an effective preconditioner to accelerate the convergence of the linear iterative solver. The key contribution of our work is to generalise the fast Poisson preconditioner, widely used for integer-order diffusion equations, so that it applies to the two-sided space fractional diffusion equation. A number of numerical experiments are presented to demonstrate the effectiveness of the preconditioner and the overall solution strategy.

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8 citations in Web of Science®

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ID Code: 58429
Item Type: Journal Article
Refereed: Yes
Keywords: two-sided fractional diffusion, fast Poisson preconditioner, nonlinear, method of lines, Jacobian-free Newton-Krylov
DOI: 10.1016/j.jcp.2013.03.029
ISSN: 0021-9991
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 2013 Elsevier
Copyright Statement: This is the author’s version of a work that was accepted for publication in Journal of Computational Physics. 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 Journal of Computational Physics, [Volume 246, (1 August 2013)] DOI: 10.1016/j.jcp.2013.03.029
Deposited On: 20 Mar 2013 01:26
Last Modified: 02 Sep 2015 04:10

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