Strang‐type preconditioners for systems of LMF‐based ODE codes

Raymond H. Chan; Michael K. Ng; Xiao Qing Jin

doi:10.1093/imanum/21.2.451

Strang‐type preconditioners for systems of LMF‐based ODE codes

Raymond H. Chan
, Michael K. Ng
, Xiao Qing Jin

Department of Mathematics

Research output: Contribution to journal › Journal article › peer-review

41 Citations (Scopus)

Abstract

We consider the solution of ordinary differential equations (ODEs) using boundary value methods. These methods require the solution of one or more unsymmetric, large and sparse linear systems. The GMRES method with the Strang‐type block‐circulant preconditioner is proposed for solving these linear systems. We show that if an A_k1,k2 ‐stable boundary value method is used for an m‐by‐m system of ODEs, then our preconditioners are invertible and all the eigenvalues of the preconditioned systems are 1 except for at most 2m(k₁ + k₂) outliers. It follows that when the GMRES method is applied to solving the preconditioned systems, the method will converge in at most 2m(k₁ + k₂) + 1 iterations. Numerical results are given to illustrate the effectiveness of our methods.

Original language	English
Pages (from-to)	451-462
Number of pages	12
Journal	IMA Journal of Numerical Analysis
Volume	21
Issue number	2
DOIs	https://doi.org/10.1093/imanum/21.2.451
Publication status	Published - 1 Apr 2001

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Cite this

@article{1445cc5f7fcc4b17badde1effb5ab834,

title = "Strang‐type preconditioners for systems of LMF‐based ODE codes",

abstract = "We consider the solution of ordinary differential equations (ODEs) using boundary value methods. These methods require the solution of one or more unsymmetric, large and sparse linear systems. The GMRES method with the Strang‐type block‐circulant preconditioner is proposed for solving these linear systems. We show that if an Ak1,k2 ‐stable boundary value method is used for an m‐by‐m system of ODEs, then our preconditioners are invertible and all the eigenvalues of the preconditioned systems are 1 except for at most 2m(k1 + k2) outliers. It follows that when the GMRES method is applied to solving the preconditioned systems, the method will converge in at most 2m(k1 + k2) + 1 iterations. Numerical results are given to illustrate the effectiveness of our methods.",

author = "Chan, \{Raymond H.\} and Ng, \{Michael K.\} and Jin, \{Xiao Qing\}",

note = "Publisher Copyright: {\textcopyright} The Institute of Mathematics and its Applications 2001. Funding Information: Research supported in part by Hong Kong Research Grants Council Grant No. CUHK 4207/97P and CUHK DAG Grant No. 2060143. Research supported in part by Hong Kong Research Grants Council Grant No. HKU 7147/99P and UK/HK Joint Research Scheme Grant No. 20009819. Research supported in part by UM Research Grant No. RG009/98-99S/JXQ/FST.",

year = "2001",

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doi = "10.1093/imanum/21.2.451",

language = "English",

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journal = "IMA Journal of Numerical Analysis",

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T1 - Strang‐type preconditioners for systems of LMF‐based ODE codes

AU - Chan, Raymond H.

AU - Ng, Michael K.

AU - Jin, Xiao Qing

N1 - Publisher Copyright: © The Institute of Mathematics and its Applications 2001. Funding Information: Research supported in part by Hong Kong Research Grants Council Grant No. CUHK 4207/97P and CUHK DAG Grant No. 2060143. Research supported in part by Hong Kong Research Grants Council Grant No. HKU 7147/99P and UK/HK Joint Research Scheme Grant No. 20009819. Research supported in part by UM Research Grant No. RG009/98-99S/JXQ/FST.

PY - 2001/4/1

Y1 - 2001/4/1

N2 - We consider the solution of ordinary differential equations (ODEs) using boundary value methods. These methods require the solution of one or more unsymmetric, large and sparse linear systems. The GMRES method with the Strang‐type block‐circulant preconditioner is proposed for solving these linear systems. We show that if an Ak1,k2 ‐stable boundary value method is used for an m‐by‐m system of ODEs, then our preconditioners are invertible and all the eigenvalues of the preconditioned systems are 1 except for at most 2m(k1 + k2) outliers. It follows that when the GMRES method is applied to solving the preconditioned systems, the method will converge in at most 2m(k1 + k2) + 1 iterations. Numerical results are given to illustrate the effectiveness of our methods.

AB - We consider the solution of ordinary differential equations (ODEs) using boundary value methods. These methods require the solution of one or more unsymmetric, large and sparse linear systems. The GMRES method with the Strang‐type block‐circulant preconditioner is proposed for solving these linear systems. We show that if an Ak1,k2 ‐stable boundary value method is used for an m‐by‐m system of ODEs, then our preconditioners are invertible and all the eigenvalues of the preconditioned systems are 1 except for at most 2m(k1 + k2) outliers. It follows that when the GMRES method is applied to solving the preconditioned systems, the method will converge in at most 2m(k1 + k2) + 1 iterations. Numerical results are given to illustrate the effectiveness of our methods.

UR - https://www.scopus.com/pages/publications/0035531729

U2 - 10.1093/imanum/21.2.451

DO - 10.1093/imanum/21.2.451

M3 - Journal article

AN - SCOPUS:0035531729

SN - 0272-4979

VL - 21

SP - 451

EP - 462

JO - IMA Journal of Numerical Analysis

JF - IMA Journal of Numerical Analysis

IS - 2

ER -

Strang‐type preconditioners for systems of LMF‐based ODE codes

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