I'm students at undergraduate school 2year
I don't have much background knowledge
I study nummerical analysis Runge - Kutta Method not a few
my text book induce Runge - Kutta midpoint method by 2 order Taylor series $T^2$ and it's approximating function $a_1f(t+\alpha_1,y+\beta_1)$
this approximation error is $O(h^2)$
since $w_{i+1}=w_i + h T^2(t_i,w_i) $ where $T^2(t,y)$ is order 2 Taylor for f(t,y). its local truncation error is also two
then replace $ T^2(t_i,w_i)$ to $ f(t_i+\frac{h}{2},w_i+\frac{h}{2}f(t_i,w_i))$ thus $w_{i+1}=w_i + hf(t_i+\frac{h}{2},w_i+\frac{h}{2}f(t_i,w_i))$
is this local truncation error also two because approximation error and local truncation error of $w_{i+1}=w_i + h T^2(t_i,w_i) $ is same for two?
is it correct?
sorry to my poor English ability