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On sinc discretization and banded preconditioning for linear third‐order ordinary differential equations
Author(s) -
Bai ZhongZhi,
Chan Raymond H.,
Ren ZhiRu
Publication year - 2011
Publication title -
numerical linear algebra with applications
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.02
H-Index - 53
eISSN - 1099-1506
pISSN - 1070-5325
DOI - 10.1002/nla.738
Subject(s) - mathematics , generalized minimal residual method , krylov subspace , coefficient matrix , ordinary differential equation , discretization , toeplitz matrix , linear system , mathematical analysis , eigenvalues and eigenvectors , sinc function , matrix (chemical analysis) , differential equation , physics , materials science , quantum mechanics , pure mathematics , composite material
Some draining or coating fluid‐flow problems and problems concerning the flow of thin films of viscous fluid with a free surface can be described by third‐order ordinary differential equations (ODEs). In this paper, we solve the boundary value problems of such equations by sinc discretization and prove that the discrete solutions converge to the true solutions of the ODEs exponentially. The discrete solution is determined by a linear system with the coefficient matrix being a combination of Toeplitz and diagonal matrices. The system can be effectively solved by Krylov subspace iteration methods, such as GMRES, preconditioned by banded matrices. We demonstrate that the eigenvalues of the preconditioned matrix are uniformly bounded within a rectangle on the complex plane independent of the size of the linear system. Numerical examples are given to illustrate the effective performance of our method. Copyright © 2010 John Wiley & Sons, Ltd.