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

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

*Corresponding author for this work

Research output: Contribution to journalArticleAcademicpeer-review

55 Citations (Scopus)

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 Grunwald 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 O(N-2) to O(N) and the computational complexity from O(N-3) to 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.

Original languageEnglish
Pages (from-to)957-985
Number of pages29
JournalJournal of scientific computing
Volume72
Issue number3
DOIs
Publication statusPublished - Sep-2017

Keywords

  • Fractional convection-diffusion equation
  • Shifted Grunwald discretization
  • Toeplitz matrix
  • Fast Fourier transform
  • Circulant preconditioner
  • Krylov subspace method
  • ADVECTION-DISPERSION EQUATION
  • NONLINEAR SOURCE-TERM
  • COMPACT EXPONENTIAL SCHEME
  • FINITE-VOLUME METHOD
  • NUMERICAL-SOLUTION
  • VARIABLE-COEFFICIENTS
  • COLLOCATION METHOD
  • HIGH-ORDER
  • MESHLESS METHOD
  • APPROXIMATIONS

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