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

Abstract

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. 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 ordinary differential equations 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.

Banded Toeplitz Preconditioners for Toeplitz Matrices from Sinc Methods

“…When p is an even number, from Theorem 2,1 we know that T^"*) (m = 2p-|-1) is a skew-symmetric Toeplitz matrix with its generating function in the Wiener class. By making use of Theorems 3,1 and 3,3 in (ii) can be used to estimate generalized Rayleigh quotients of the Hermitian and skew-Hermitian parts of the coefficient matrices obtaining from the sine discretization for linear ordinary or partial differential equations, as the Hermitian parts of the coefficient matrices are combinations of Tt"") (m is even) and diagonal matrices D; see [1][2][3]. Because preconditioners for the coefficient matrices can be formed as the same structure as the coefficient matrices, we can also employ Theorem 4.1 (ii) to estimate generalized Rayleigh quotients of the Hermitian and skew-Hermitian parts of the corresponding structured preconditioners.…”

“…In particular, when the sine method is applied to discretize the linear ordinary and partial differential equations, we can often obtain systems of linear equations whose coefficient matrices are combinations of Toeplitz and diagonal matrices; see [1][2][3]10,14]. Hence, it 534 Z.-R. REN is a basic requirement to discuss the algebraic properties of these Toeplitz matrices, construct their efiPective preconditioners, and derive tight eigenvalue bounds for the corresponding preconditioned matrices.…”

The Explicit Inverses of CUPL-Toeplitz and CUPL-Hankel Matrices

“…As is well known, Toeplitz matrices family are also structured matrices family and have important applications in various disciplines including the elliptic Dirichlet-periodic boundary value problems [1], sinc discretizations of partial and ordinary differential equations [2][3][4][5][6][7], signal processing [8], numerical analysis [8], system theory [8], etc.…”

On order-reducible sinc discretizations and block-diagonal preconditioning methods for linear third-order ordinary differential equations

“…The coefficient matrix of the resulted linear system is a combination of Toeplitz and diagonal matrices. In [17], the authors constructed a banded preconditioning matrix and employed the corresponding preconditioned Krylov subspace methods to iteratively solve such a discretized linear system. However, as the highest term of the ODE (1) is of order three, the discretized linear system is highly ill-conditioned as its coefficient matrix has a strongly dominant skew-symmetric part.…”

“…However, as the highest term of the ODE (1) is of order three, the discretized linear system is highly ill-conditioned as its coefficient matrix has a strongly dominant skew-symmetric part. As a result, in actual implementations there is considerable difficulty in iteratively computing the discrete solution.To overcome the shortcoming of the direct discretization method in [17], in this paper we introduce a variable substitution that transforms the third-order ODE (1) into a system of two secondorder ODEs. Then, we apply the approach similar to the one used in [17] to solve the resulting system.…”