Algorithmic Realization of an Exact Three-Point Difference Scheme for the Sturm–Liouville Problem

We develop a new algorithmic implementation of the exact three-point difference schemes on a nonuniform grid for the Sturm–Liouville problem. It is shown that, in order to find the coefficients of exact scheme for an arbitrary node of the grid, it is necessary to solve two auxiliary Cauchy problems for second-order linear ordinary differential equations: one problem on the interval [ x j −1 , x j ] (forward) and one problem on the interval [ x j , x j +1 ] (backward). We prove the theorem on the coefficient stability of the exact three-point difference scheme.