A τ-preconditioner for a non-symmetric linear system arising from multi-dimensional Riemann-Liouville fractional diffusion equation

Xue lei Lin, Xin Huang, Michael K. Ng*, Hai Wei Sun

*Corresponding author for this work

Research output: Contribution to journalJournal articlepeer-review

4 Citations (Scopus)

Abstract

In this paper, we study a τ-preconditioner for non-symmetric linear system arising from a steady-state multi-dimensional Riemann-Liouville (RL) fractional diffusion equation. The generalized minimal residual (GMRES) method is applied to solve the preconditioned linear system. Theoretically, we show that the GMRES solver for the preconditioned linear system has a convergence rate independent of discretization stepsizes. To the best of our knowledge, this is the first iterative solver with stepsize-independent convergence rate for the non-symmetric linear system. The proposed τ-preconditioner is diagonalizable by the sine transform matrix, thanks to which the matrix-vector multiplication in each iteration step can be fast implemented by the fast sine transform (FST). Hence, the total operation cost of the proposed solver for the non-symmetric problem is linearithmic. Numerical results are reported to show the efficiency of the proposed preconditioner.

Original languageEnglish
Pages (from-to)795-813
Number of pages19
JournalNumerical Algorithms
Volume92
Issue number1
Early online date1 Jul 2022
DOIs
Publication statusPublished - Jan 2023

Scopus Subject Areas

  • Applied Mathematics

User-Defined Keywords

  • Convergence of GMRES
  • Fractional diffusion equation
  • Non-symmetric linear system
  • Preconditioning

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