In this paper, we propose a fast block $\alpha$-circulant preconditioner for solving the nonsymmetric linear system arising from an all-at-once implicit discretization scheme in time for the wave equation. As a generalization of the well-known block circulant preconditioning technique, the proposed block $\alpha$-circulant preconditioner can also be efficiently inverted in a parallel-in-time manner. The complex eigenvalues of the preconditioned matrix are fully derived in explicit expression and its diagonalizability is also shown. Furthermore, a mesh-independent convergence rate of the preconditioned GMRES method is derived under certain conditions. Building upon the proposed preconditioner, two stationary iterative methods with uniform asymptotic convergence rates were also presented. The extension of our preconditioner within a simplified Newton iteration to the nonlinear wave equation is also discussed. Both linear and nonlinear numerical examples are given to illustrate the promising performance of our proposed block $\alpha$-circulant preconditioner and also validate our theoretical results on convergence analysis.


  1. all-at-once scheme
  2. wave equations
  3. $\alpha$-circulant preconditioner
  4. parallel-in-time (PinT)
  5. diagonalization

MSC codes

  1. 65M55
  2. 65M12
  3. 65M15
  4. 65Y05

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Published In

cover image SIAM Journal on Matrix Analysis and Applications
SIAM Journal on Matrix Analysis and Applications
Pages: 1912 - 1943
ISSN (online): 1095-7162


Submitted: 30 December 2019
Accepted: 14 September 2020
Published online: 10 December 2020


  1. all-at-once scheme
  2. wave equations
  3. $\alpha$-circulant preconditioner
  4. parallel-in-time (PinT)
  5. diagonalization

MSC codes

  1. 65M55
  2. 65M12
  3. 65M15
  4. 65Y05



Funding Information

NSF of China : 11771313
NSF of Sichuan : 2018JY0469

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