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Journal of Scientific Computing
Erschienen in: Journal of Scientific Computing 3/2017

17.02.2017

Fast Iterative Method with a Second-Order Implicit Difference Scheme for Time-Space Fractional Convection–Diffusion Equation

verfasst von: Xian-Ming Gu, Ting-Zhu Huang, Cui-Cui Ji, Bruno Carpentieri, Anatoly A. Alikhanov

Erschienen in: Journal of Scientific Computing | Ausgabe 3/2017

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Abstract

In this paper we intend to establish fast numerical approaches to solve a class of initial-boundary problem of time-space fractional convection–diffusion equations. We present a new unconditionally stable implicit difference method, which is derived from the weighted and shifted Grünwald formula, and converges with the second-order accuracy in both time and space variables. Then, we show that the discretizations lead to Toeplitz-like systems of linear equations that can be efficiently solved by Krylov subspace solvers with suitable circulant preconditioners. Each time level of these methods reduces the memory requirement of the proposed implicit difference scheme from \({\mathcal {O}}(N^2)\) to \({\mathcal {O}}(N)\) and the computational complexity from \({\mathcal {O}}(N^3)\) to \({\mathcal {O}}(N\log N)\) in each iterative step, where N is the number of grid nodes. Extensive numerical examples are reported to support our theoretical findings and show the utility of these methods over traditional direct solvers of the implicit difference method, in terms of computational cost and memory requirements.
Vorheriger Artikel An Adaptive Finite Element Method for the Wave Scattering with Transparent Boundary Condition
Nächster Artikel Adaptive WENO Methods Based on Radial Basis Function Reconstruction

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Fußnoten
1
In this case, it should mention that we only need to solve three nonsymmetric Toeplitz systems, i.e., equations with the form like (3.5) and in Step 4 of Algorithm 3, for implementing the whole for loop.
 
2
For the sake of clarity, here we do not list the number of iterations required for solving those three linear systems one by one.
 
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Metadaten
Titel
Fast Iterative Method with a Second-Order Implicit Difference Scheme for Time-Space Fractional Convection–Diffusion Equation
verfasst von
Xian-Ming Gu
Ting-Zhu Huang
Cui-Cui Ji
Bruno Carpentieri
Anatoly A. Alikhanov
Publikationsdatum
17.02.2017
Verlag
Springer US
Erschienen in
Journal of Scientific Computing / Ausgabe 3/2017
Print ISSN: 0885-7474
Elektronische ISSN: 1573-7691
DOI
https://doi.org/10.1007/s10915-017-0388-9

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