An explicit expression of ordinary difference schemes for differential equations by the moved node method

This article discusses the issue of the possibility of calculating the approximation error. When replacing differential equations with discrete ones, one of the key issues is the closeness of the discrete solution to the exact solution. For the difference solution of the problem, a grid area is formed. The discrete solution is determined at the nodal points. Traditionally, in questions of replacing a differential equation with a descriptive one, one usually indicates the degree of approximation of the O(hp) type. Here h is the grid step. However, it is possible to calculate the approximation error at nodal points based on the method of moving nodes. The method of moving nodes allows obtaining an approximate analytical expression. On the basis of the approximate form, it is possible to calculate the approximation error. On the other hand, at each node one can construct a differential analog of the difference equation. Using simple examples, the calculation of approximation errors is demonstrated and schemes of the collocation type are constructed.

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