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.

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For the solving Sturm-Liouville problem, three-point difference schemes of high order of accuracy on a nonuniform grid are constructed. It is shown that the coefficients of these schemes are expressed in terms of solutions of two auxiliary initial value problems. An estimate of the accuracy of three-point difference schemes is obtained and an iterative Newton method is proposed to determine their solution. Numerical experiments confirm theoretical conclusions.