Banded preconditioners for Riesz space fractional diffusion equations
type: Item
instance of: scholarly article
title: Banded preconditioners for Riesz space fractional diffusion equations
zbMATH Open document ID: 1475.65079
author: Zi-Hang She
author: Cheng-Xue Lao
author: Hong Yang
author: Fu-Rong Lin
published in: Journal of Scientific Computing
publication date: 2021-05-28T00:00:00Z
review text: In this paper, numerical methods for Toeplitz-like linear systems arising from the one- and two-dimensional Riesz space fractional diffusion equations are considered. Crank-Nicolson technique is applied to discretize the temporal derivative and apply certain difference operator to discretize the space fractional derivatives. For the one-dimensional problem, the corresponding coefficient matrix is the sum of an identity matrix and a product of a diagonal matrix and a symmetric Toeplitz matrix. They transform the linear systems to symmetric linear systems and introduce symmetric banded preconditioners. They prove that under mild assumptions, the eigenvalues of the preconditioned matrix are bounded above and below by positive constants. In particular, the lower bound of the eigenvalues is equal to 1 when the banded preconditioner with diagonal compensation is applied. The preconditioned conjugate gradient method is applied to solve relevant linear systems. Numerical results are presented to verify the theoretical results about the preconditioned matrices and to illustrate the efficiency of the proposed preconditioners. In my opinion, this work is interesting and makes some progress in fast algorithm for solving fractional partial differential equation.
reviewed by: Qifeng Zhang
Mathematics Subject Classification ID: 65M06
Mathematics Subject Classification ID: 65N06
Mathematics Subject Classification ID: 35R11
Mathematics Subject Classification ID: 26A33
Mathematics Subject Classification ID: 65F08
Mathematics Subject Classification ID: 65F10
Mathematics Subject Classification ID: 65F35
Mathematics Subject Classification ID: 15B05
Mathematics Subject Classification ID: 34A08
zbMATH DE Number: 7352176
zbMATH Keywords: Riesz space fractional diffusion equation
zbMATH Keywords: PCG methods
zbMATH Keywords: banded preconditioner
zbMATH Keywords: condition number
MaRDI profile type: Publication
full work available at URL: s10915-020-01398-4
OpenAlex ID: W3125886675
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DOI: 10.1007/S10915-020-01398-4
description: scientific article; zbMATH DE number 7352176