Midpoint Method - Derivation of The Midpoint Method

Derivation of The Midpoint Method

The midpoint method is a refinement of the Euler's method

and is derived in a similar manner. The key to deriving Euler's method is the approximate equality

which is obtained from the slope formula

and keeping in mind that

For the midpoint method, one replaces (3) with the more accurate

when instead of (2) we find

One cannot use this equation to find as one does not know at The solution is then to use a Taylor series expansion exactly as if using the Euler method to solve for :

which, when plugged in (4), gives us

and the midpoint method (1).