A New Approach to Calculating the Approximation Error of Ordinary Differential Equations

It is known that the degree of proximity of the solutions of a differential equation and its difference analogue is estimated in the form O(hp) (h is the grid step, p is the order of approximation). In this case, the order of approximation is derived on the basis of the difference equation using the Taylor series.

Using the moving node method, it is possible to obtain an approximate analytical expression for the approximation error. Based on the approximate form of the approximation error, the approximation error can be calculated.

The analytical form of the approximation form allows for the efficient calculation of this error. However, the property of this error allows you to create new and improved circuits. Furthermore, based on these types of errors, a differential analogue of the difference equation can be created, yielding an exact approximation.