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

Fast Iterative Method with a Second-Order Implicit Difference Scheme for Time-Space Fractional... 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 $${\mathcal {O}}(N^2)$$ O ( N 2 ) to $${\mathcal {O}}(N)$$ O ( N ) and the computational complexity from $${\mathcal {O}}(N^3)$$ O ( N 3 ) to $${\mathcal {O}}(N\log N)$$ 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. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Journal of Scientific Computing Springer Journals

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

, Volume 72 (3) – Feb 17, 2017
29 pages

/lp/springer_journal/fast-iterative-method-with-a-second-order-implicit-difference-scheme-Fb2ybMtkP6
Publisher
Springer US
Subject
Mathematics; Algorithms; Computational Mathematics and Numerical Analysis; Mathematical and Computational Engineering; Theoretical, Mathematical and Computational Physics
ISSN
0885-7474
eISSN
1573-7691
D.O.I.
10.1007/s10915-017-0388-9
Publisher site
See Article on Publisher Site

### 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 $${\mathcal {O}}(N^2)$$ O ( N 2 ) to $${\mathcal {O}}(N)$$ O ( N ) and the computational complexity from $${\mathcal {O}}(N^3)$$ O ( N 3 ) to $${\mathcal {O}}(N\log N)$$ 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.

### Journal

Journal of Scientific ComputingSpringer Journals

Published: Feb 17, 2017

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