For a given ODE $y'(t)=f(t,y)$, Euler's method is $ y(t+h)=y(t)+hf(t,y(t)) + O(h^2) $
It is said that by using Richardson extrapolation, we can improve it to $ y(t+h)=y(t)+hf(t+\frac{h}{2},y(t)+\frac{h}{2}f(t,y)) + O(h^3) $
But I don't know how to deduce it.
To use Richardson extrapolation, we need a quantity of interest $A$ and its approximation $A(h)$, what should them be?
BTW, I know we can use the formula $y'(x+h/2)=\frac{y(x+h)-y(x)}{h}+O(h^3)$ to get improved Euler's method. But here I wonder how to get it from Richardson's method?