Mariarosa Mazza,
Stefano Serra-Capizzano,
Muhammad Usman
S. 499 - 513
doi:
10.1553/etna_vol54s499
Verlag der Österreichischen Akademie der Wissenschaften
doi:
10.1553/etna_vol54s499
Abstract:
In this work, we examine the numerical solution of a 1D distributed-order space-fractional diffusion equation. Discretizing the given problem by means of an implicit finite difference scheme based on the shifted Grünwald-Letnikov formula, the resulting linear systems show a Toeplitz structure. Then, by using well-known spectral tools for Toeplitz sequences, we determine the corresponding symbol describing its asymptotic eigenvalue distribution as the matrix size diverges. The spectral analysis is performed under different assumptions with the aim of estimating the intrinsic asymptotic ill-conditioning of the involved matrices. The obtained results suggest to precondition the involved linear systems with either a Laplacian-like preconditioner or with more general τ-preconditioners. Due to the symmetric positive definite nature of the coefficient matrices, we opt for the preconditioned conjugate gradient method, and we compare the performances of our proposal with a Strang circulant alternative given in the literature.
fractional diffusion equations, Toeplitz matrices, spectral distribution, preconditioning
Published Online:
2021/08/09 12:02:09
Object Identifier:
0xc1aa5576 0x003cb5c1
Rights:All rights reserved.For questions regarding copyright and copies please contact us by email.
Electronic Transactions on Numerical Analysis (ETNA) is an electronic journal for the publication of significant new developments in numerical analysis and scientific computing. Papers of the highest quality that deal with the analysis of algorithms for the solution of continuous models and numerical linear algebra are appropriate for ETNA, as are papers of similar quality that discuss implementation and performance of such algorithms. New algorithms for current or new computer architectures are appropriate provided that they are numerically sound. However, the focus of the publication should be on the algorithm rather than on the architecture. The journal is published by the Kent State University Library in conjunction with the Institute of Computational Mathematics at Kent State University, and in cooperation with the Johann Radon Institute for Computational and Applied Mathematics of the Austrian Academy of Sciences (RICAM). Reviews of all ETNA papers appear in Mathematical Reviews and Zentralblatt für Mathematik. Reference information for ETNA papers also appears in the expanded Science Citation Index. ETNA is registered with the Library of Congress and has ISSN 1068-9613.
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