Block-triangular preconditioning methods for linear third-order ordinary differential equations based on reduced-order sinc discretizations

Abstract

By applying the reduced-order sinc discretization to the two-point boundary value problem of a linear third-order ordinary differential equation, we can obtain a block two-by-two system of linear equations, with each block of its coefficient matrix being a combination of Toeplitz and diagonal matrices. This class of linear systems can be effectively solved by Krylov subspace iteration methods such as GMRES and BiCGSTAB. We construct block-triangular preconditioning matrices to accelerate the convergence rates of the Krylov subspace iteration methods, and demonstrate that the eigenvalues of certain approximations to the preconditioned matrices are uniformly bounded within a rectangle, being independent of the size of the discrete linear system, on the complex plane. In addition, we use numerical examples to show the effectiveness of the proposed preconditioning methods.