Euler Method - Derivation

Derivation

The Euler method can be derived in a number of ways. Firstly, there is the geometrical description mentioned above.

Another possibility is to consider the Taylor expansion of the function around :

The differential equation states that . If this is substituted in the Taylor expansion and the quadratic and higher-order terms are ignored, the Euler method arises. The Taylor expansion is used below to analyze the error committed by the Euler method, and it can be extended to produce Runge–Kutta methods.

A closely related derivation is to substitute the forward finite difference formula for the derivative,

in the differential equation . Again, this yields the Euler method. A similar computation leads to the midpoint rule and the backward Euler method.

Finally, one can integrate the differential equation from to and apply the fundamental theorem of calculus to get:

Now approximate the integral by the left-hand rectangle method (with only one rectangle):

Combining both equations, one finds again the Euler method. This line of thought can be continued to arrive at various linear multistep methods.