Symbol-based preconditioning for Riesz distributed-order space-fractional diffusion equations

Electronic Transactions on Numerical Analysis

article

Mariarosa Mazza¹  Stefano Serra-Capizzano¹  Muhammad Usman² 

[1] Department of Humanities and Innovation, University of Insubria;Department of Science and High Technology, University of Insubria

关键词: fractional diffusion equations;    Toeplitz matrices;    spectral distribution;    preconditioning;   

DOI  :  10.1553/etna_vol54s499

学科分类：数学（综合）

来源: Kent State University * Institute of Computational Mathematics

【 摘 要 】

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 $\tau$-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.

【 授权许可】

Unknown