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Block-triangular preconditioning methods for linear third-order ordinary differential equations based on reduced-order sinc discretizations

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Abstract

By applying the reduced-order sinc discretization to the two-point boundary value problem of a linear third-order ordinary differential equation, we can obtain a block two-by-two system of linear equations, with each block of its coefficient matrix being a combination of Toeplitz and diagonal matrices. This class of linear systems can be effectively solved by Krylov subspace iteration methods such as GMRES and BiCGSTAB. We construct block-triangular preconditioning matrices to accelerate the convergence rates of the Krylov subspace iteration methods, and demonstrate that the eigenvalues of certain approximations to the preconditioned matrices are uniformly bounded within a rectangle, being independent of the size of the discrete linear system, on the complex plane. In addition, we use numerical examples to show the effectiveness of the proposed preconditioning methods.

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Authors and Affiliations

  1. State Key Laboratory of Scientific/Engineering Computing, Institute of Computational Mathematics and Scientific/Engineering Computing, Academy of Mathematics and Systems Science, Chinese Academy of Sciences, P.O. Box 2719, Beijing, 100190, People’s Republic of China

    Zhong-Zhi Bai & Zhi-Ru Ren

Authors
  1. Zhong-Zhi Bai
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  2. Zhi-Ru Ren
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Correspondence to Zhong-Zhi Bai.

Additional information

Supported by The Hundred Talent Project of Chinese Academy of Sciences, The National Basic Research Program (No. 2011CB309703), The National Natural Science Foundation (No. 91118001) and The National Natural Science Foundation for Creative Research Groups (No. 11021101), People’s Republic of China.

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Bai, ZZ., Ren, ZR. Block-triangular preconditioning methods for linear third-order ordinary differential equations based on reduced-order sinc discretizations. Japan J. Indust. Appl. Math. 30, 511–527 (2013). https://doi.org/10.1007/s13160-013-0112-6

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  • DOI: https://doi.org/10.1007/s13160-013-0112-6

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