Efficient preconditioner of one-sided space fractional diffusion equation

Xue Lei Lin*, Kwok Po NG, Hai Wei Sun

*Corresponding author for this work

Research output: Contribution to journalJournal articlepeer-review

22 Citations (Scopus)


In this paper, we propose an efficient preconditioner for the linear systems arising from the one-sided space fractional diffusion equation with variable coefficients. The shifted Gru ¨ nwald formula is employed to discretize the one-sided Riemann–Liouville fractional derivative. The matrix structure of resulting linear systems is Toeplitz-like, which is a summation of an identity matrix and a diagonal-times-nonsymmetric-Toeplitz matrix. A diagonal-times-nonsymmetric-Toeplitz preconditioner is proposed to reduce the condition number of the Toeplitz-like matrix, where the diagonal part comes from the variable coefficients and the nonsymmetric Toeplitz part comes from the Riemann–Liouville derivative. Theoretically, we show that the condition number of the preconditioned matrix is uniformly bounded by a constant independent of discretization step-sizes under certain assumptions on the coefficient function. Due to the uniformly bounded condition number, the Krylov subspace method for the preconditioned linear systems converges linearly and independently on discretization step-sizes. Numerical results are reported to show the efficiency of the proposed preconditioner and to demonstrate its superiority over other tested preconditioners.

Original languageEnglish
Pages (from-to)729-748
Number of pages20
JournalBIT Numerical Mathematics
Issue number3
Publication statusPublished - 1 Sept 2018

Scopus Subject Areas

  • Software
  • Computer Networks and Communications
  • Computational Mathematics
  • Applied Mathematics

User-Defined Keywords

  • One-sided space-fractional derivative
  • Preconditioning
  • Toeplitz-like matrix
  • Variable diffusion coefficients


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