TY - JOUR
T1 - On sinc discretization and banded preconditioning for linear third-order ordinary differential equations
AU - Bai, Zhong-Zhi
AU - Chan, Raymond H.
AU - Ren, Zhi-Ru
PY - 2011/5
Y1 - 2011/5
N2 - 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.
AB - 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.
KW - Banded preconditioning
KW - Convergence analysis
KW - Krylov subspace methods
KW - Sinc-collocation discretization
KW - Sinc-Galerkin discretization
KW - Third-order ordinary differential equation
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U2 - 10.1002/nla.738
DO - 10.1002/nla.738
M3 - RGC 21 - Publication in refereed journal
VL - 18
SP - 471
EP - 497
JO - Numerical Linear Algebra with Applications
JF - Numerical Linear Algebra with Applications
SN - 1070-5325
IS - 3
ER -