New Algorithmic Implementation of Exact Three-Point Difference Schemes for Systems of Nonlinear Ordinary Differential Equations of the Second Order

We propose and justify three-point difference schemes of higher orders of accuracy on a nonuniform grid for systems of nonlinear ordinary differential equations of the second order with derivative on the right-hand side and boundary conditions of the first kind. We construct a new approximation of the derivative of solution to the boundary value problem at the nodes of the grid, prove the existence and uniqueness of the solution, and establish the order of accuracy of the difference schemes. We also develop an iterative Newton-type method aimed at finding the solutions of these schemes and propose an algorithm for the automatic selection of grid points guaranteeing the attainment of the required accuracy. In addition, we present numerical examples, which confirm the efficiency and reliability of the proposed algorithm.