Analysis of Difference Schemes of Two-Point Boundary Value Problems using the Method of Moving Nodes

This article addresses the calculation of approximation errors in numerical methods for solving differential equations. A fundamental challenge when replacing differential equations with discrete representations is ensuring that the discrete solution closely approximates the exact solution. To tackle this, a grid area is established for the difference solution, with discrete solutions evaluated at specific nodal points. Traditionally, the degree of approximation in this context is expressed using the notation, where h represents the grid step and p indicates the order of accuracy. A significant advancement in this area is the application of the moving nodes method, which enables the calculation of approximation errors at these nodal points. This method allows researchers to derive an approximate analytical expression for the discrete solution, which serves as a foundation for calculating the approximation error.