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.

No abstract

No abstract

No abstract

Numerical methods for solving the initial value problem for ordinary differential equations are proposed. Embedded methods of order of accuracy 2(1), 3(2) and 4(3) are constructed. To estimate the local error, two-sided calculation formulas were used, which give estimates of the main terms of the error without additional calculations of the right-hand side of the differential equation, which favorably distinguishes them from traditional two-sided methods of the Runge- Kutta type.