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. We discretize this order-reduced system of ordinary differential equations 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 ordinary differential equations. The coefficient matrix of the linear system is of block two-by-two structure and each of its blocks is a combination of Toeplitz ∗Supported by The Hundred Talent Project of Chinese Academy of Sciences, and The National Basic Research Program (No. 2011CB309703), P.R. China; and by Hong Kong Research Grants Council Grant (No. 400510) and CUHK DAG 2060408.