A separable preconditioner for time-space fractional Caputo-Riesz diffusion equations

Xuelei Lin*, Kwok Po NG, Haiwei Sun

*Corresponding author for this work

Research output: Contribution to journalJournal articlepeer-review

15 Citations (Scopus)


In this paper, we study linear systems arising from time-space fractional Caputo-Riesz diffusion equations with time-dependent diffusion coefficients. The coefficient matrix is a summation of a block-lower-triangular-Toeplitz matrix (temporal component) and a block-diagonal-with-diagonal-times-Toeplitz-block matrix (spatial component). The main aim of this paper is to propose separable preconditioners for solving these linear systems, where a block ε-circulant preconditioner is used for the temporal component, while a block diagonal approximation is used for the spatial variable. The resulting preconditioner can be block-diagonalized in the temporal domain. Furthermore, the fast solvers can be employed to solve smaller linear systems in the spatial domain. Theoretically, we show that if the diffusion coefficient (temporal-dependent or spatial-dependent only) function is smooth enough, the singular values of the preconditioned matrix are bounded independent of discretization parameters. Numerical examples are tested to show the performance of proposed preconditioner.

Original languageEnglish
Pages (from-to)827-853
Number of pages27
JournalNumerical Mathematics
Issue number4
Early online dateJun 2018
Publication statusPublished - Nov 2018

Scopus Subject Areas

  • Modelling and Simulation
  • Control and Optimization
  • Computational Mathematics
  • Applied Mathematics

User-Defined Keywords

  • Block lower triangular
  • Block ε-circulant preconditioner
  • Diagonalization
  • Separable
  • Time-space fractional diffusion equations
  • Toeplitz-like matrix


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