Numerical Analysis/Order of RK methods/Implicit RK2 on an Autonomous ODE

We consider an autonomous initial value ODE

Applying the Tradezoidal rule gives the implicit Runge-Kutta method

We will show that ($$) is second order.

Expanding the true solution $$y(t_n+h)$$ about $$t_n$$ using Taylor series, we have
 * $$ y(t_n+h)=y(t_n)+hy'(t_n)+\frac{h^2}{2}y(t_n)+\frac{h^3}{3!}y'(t_n)+O(h^4)\,.$$

Since $$y(t)$$ satisfies ($$), we can substitute $$y'(t)=f(y(t))$$ and obtain

In ($$) we can assume $$y_n=y(t_n)$$ since that is the previous data. Subtracting ($$) from ($$) gives us the local truncation error

In order to cancel more terms we need to expand $$f(y_{n+1})$$. However, $$y_{n+1}\not=y(t_n+h)$$ so we cannot do a regular Taylor expansion. Instead we can plug ($$) back into $$f$$ and then do a Taylor expansion to obtain

Substituting ($$) into ($$) yields

This substitution was productive since the $$h$$ terms canceled. We can do this trick again, but this time only need ($$) up to $$O(h^2)$$ since everything will be multiplied by at least $$h^2$$ and this can go into the $$O(h^4)$$. Substituting ($$) in for the first occurance of $$f(y_{n+1})$$ in ($$) yields

This substitution was productive since the $$h^2$$ terms canceled. We can do this again, now truncating ($$) at $$O(h)$$. Substituting ($$) into ($$) yields
 * $$ \frac{-h^3}{12}\left\{f''(y(t_n))(f(y(t_n)))^2 +(f'(y(t_n)))^2f(y(t_n))\right\}+O(h^4)\,.$$

Since the $$h^3$$ term does not cancel, we have shown that the local truncation error is $$O(h^3)$$ and thus the method is order 2.