Preconditioning for symmetric positive definite systems in balanced fractional diffusion equations

In this paper, we study the finite volume discretization method for balanced fractional diffusion equations where the fractional differential operators are comprised of both Riemann-Liouville and Caputo fractional derivatives. The main advantage of this approach is that new symmetric positive defini...

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Bibliographic Details
Published in:Numerische Mathematik 2021-03, Vol.147 (3), p.651-677
Main Authors: Fang, Zhi-Wei, Lin, Xue-Lei, Ng, Michael K., Sun, Hai-Wei
Format: Article
Language:English
Subjects:
Conjugate gradient method
Conjugates
Differential equations
Diffusion
Discretization
Linear systems
Mathematical analysis
Mathematical and Computational Engineering
Mathematical and Computational Physics
Mathematical Methods in Physics
Mathematics
Mathematics and Statistics
Numerical Analysis
Numerical and Computational Physics
Numerical methods
Operators (mathematics)
Preconditioning
Simulation
Theoretical
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Summary:In this paper, we study the finite volume discretization method for balanced fractional diffusion equations where the fractional differential operators are comprised of both Riemann-Liouville and Caputo fractional derivatives. The main advantage of this approach is that new symmetric positive definite Toeplitz-like linear systems can be constructed for solving balanced fractional diffusion equations when diffusion functions are non-constant. It is different from non-symmetric Toeplitz-like linear systems usually obtained by existing numerical methods for fractional diffusion equations. The preconditioned conjugate gradient method with circulant and banded preconditioners can be applied to solve the proposed symmetric positive definite Toeplitz-like linear systems. Numerical examples, for both of one- and two- dimensional cases, are given to demonstrate the good accuracy of the finite volume discretization method and the fast convergence of the preconditioned conjugate gradient method.
ISSN:0029-599X
0945-3245
DOI:10.1007/s00211-021-01175-x