A preconditioned Lanczos method for space-fractional reaction-diffusion equations on two-dimensional unstructured meshes

Yang, Qianqian, Turner, Ian, Moroney, Timothy J., & Liu, Fawang (2012) A preconditioned Lanczos method for space-fractional reaction-diffusion equations on two-dimensional unstructured meshes. In Gu, YuanTong (Ed.) 4th International Conference on Computational Methods (ICCM2012), 25-28 November 2012, Crowne Plaza, Gold Coast, QLD.

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Abstract

We consider a two-dimensional space-fractional reaction diffusion equation with a fractional Laplacian operator and homogeneous Neumann boundary conditions. The finite volume method is used with the matrix transfer technique of Ilić et al. (2006) to discretise in space, yielding a system of equations that requires the action of a matrix function to solve at each timestep. Rather than form this matrix function explicitly, we use Krylov subspace techniques to approximate the action of this matrix function. Specifically, we apply the Lanczos method, after a suitable transformation of the problem to recover symmetry. To improve the convergence of this method, we utilise a preconditioner that deflates the smallest eigenvalues from the spectrum. We demonstrate the efficiency of our approach for a fractional Fisher’s equation on the unit disk.

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ID Code: 58432
Item Type: Conference Paper
Refereed: Yes
Keywords: Fractional Laplacian, matrix transfer technique, matrix function, Lanczos method, Krylov subspace, preconditioner, deflation, fractional Fisher’s equation
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
Current > QUT Faculties and Divisions > Science & Engineering Faculty
Copyright Owner: Copyright 2012 [please consult the author]
Deposited On: 19 Mar 2013 01:06
Last Modified: 27 Mar 2013 22:08

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