I'm studying the local truncation error of each Heun's, Midpoint, and Ralston's methods. For Heun's method, I found a material in here. However, I don't get how the following is derived. $$ f(t+h, y(t+h)) = f(t+t, y(t)+hf(t,y(t)) + \frac 1 2 f_2(\tau_3, y(\tau_3))y^{(2)}(\tau_3)h^2 $$ where $$ f_2(\tau_3,y(\tau_3)) = \left.\frac{\partial}{\partial y}f(\tau_3,y)\right|_{y=y(\tau_3)} $$ I can understand the $\tau_3$ comes from the mean-value theorem, but I cannot understand the calculus itself.
I also could not find the detailed explanation of error analyses for Midpoint and Ralston's method. Where can I find them? Please.
This is just Taylor's formula... First in one variable: $$ y(t+h) = y(t) + h y'(t) + \frac{h^2}{2}y''(\xi)= y(t) + h f(t,y) + \frac{h^2}{2}y''(\xi_t) $$
Then is two variables,
\begin{align*} f(t+h, y(t+h)) = & f\left(t+h, y(t) + h f(t, y(t)) + \frac{h^2}2 h^2y''(\xi_t)\right)\\ =& f\Bigl(t+h, y(t)+hf(t,y(t))\Bigr) + \frac{h^2}{2} y''(\xi_t) f'_y(t,y(\tilde \xi_t)) \end{align*}