Let's consider the differential equation (1) with an initial condition
Substituting into the equation (1), we’ll receive value of a derivative in a point :
At small the following expression takes place:
Designating , let’s rewrite the last equality in the form of: (2) Accepting now for a new initial point, precisely also we’ll receive
In the general case we’ll have: (3)
It also is
More exact is
and then the recalculation gives us too approached, but more exact value (correction): (4) Actually the recalculation allows to consider, though approximately, a change of a derivative on a step of integration as its values in the beginning and in the end of a step (Fig. 1) are considered, and then their average is chosen. Euler's method with recalculation (4) is in essence the 2-nd order Runge-Kutta method [2]. This will become obvious of the further.
Fig. 1. The geometric representation of Euler’s method with recalculation. |

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>> Numerical Methods
>> Ordinary Differential Equations
>> Euler’s method