On sinc discretization and banded preconditioning for linear third-order ordinary differential equations

Research output: Journal Publications and Reviews (RGC: 21, 22, 62)21_Publication in refereed journalpeer-review

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Original languageEnglish
Pages (from-to)471-497
Journal / PublicationNumerical Linear Algebra with Applications
Issue number3
Online published13 Apr 2011
Publication statusPublished - May 2011
Externally publishedYes


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.  

Research Area(s)

  • Banded preconditioning, Convergence analysis, Krylov subspace methods, Sinc-collocation discretization, Sinc-Galerkin discretization, Third-order ordinary differential equation