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

By introducing a variable substitution, we transform the two-point boundary value problem of a third-order ordinary differential equation into a system of two second-order ordinary differential equations (ODEs). We discretize this order-reduced system of ODEs by both sinc-collocation and sinc-Galerkin methods, and average these two discretized linear systems to obtain the target system of linear equations. We prove that the discrete solution resulting from the linear system converges exponentially to the true solution of the order-reduced system of ODEs. The coefficient matrix of the linear system is of block two-by-two structure, and each of its blocks is a combination of Toeplitz and diagonal matrices. Because of its algebraic properties and matrix structures, the linear system can be effectively solved by Krylov subspace iteration methods such as GMRES preconditioned by block-diagonal matrices. We demonstrate that the eigenvalues of certain approximation to the preconditioned matrix are uniformly bounded within a rectangle on the complex plane independent of the size of the discretized linear system, and we use numerical examples to illustrate the feasibility and effectiveness of this new approach.