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

We consider an autonomous initial value ODE Template:NumBlk Applying the Tradezoidal rule gives the implicit Runge-Kutta method Template:NumBlk We will show that (Template:EquationNote) is second order.

Expanding the true solution $y(t_n+h)$ about $t_n$ using Taylor series, we have

$y(t_n+h)=y(t_n)+h{y}^{\prime }(t_n)+\frac{h^2}{2}{y}^{″}(t_n)+\frac{h^3}{3!}{y}^{‴}(t_n)+O(h^4).$

Since $y(t)$ satisfies (Template:EquationNote), we can substitute $y^{\prime }(t)=f(y(t))$ and obtain Template:NumBlk In (Template:EquationNote) we can assume $y_n=y(t_n)$ since that is the previous data. Subtracting (Template:EquationNote) from (Template:EquationNote) gives us the local truncation error Template:NumBlk In order to cancel more terms we need to expand $f(y_{n+1})$. However, $y_{n+1}=y(t_n+h)$ so we cannot do a regular Taylor expansion. Instead we can plug (Template:EquationNote) back into $f$ and then do a Taylor expansion to obtain Template:NumBlk Substituting (Template:EquationNote) into (Template:EquationNote) yields Template:NumBlk This substitution was productive since the $h$ terms canceled. We can do this trick again, but this time only need (Template:EquationNote) 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 (Template:EquationNote) in for the first occurance of $f(y_{n+1})$ in (Template:EquationNote) yields Template:NumBlk This substitution was productive since the $h^2$ terms canceled. We can do this again, now truncating (Template:EquationNote) at $O(h)$. Substituting (Template:EquationNote) into (Template:EquationNote) yields

$\frac{-h^3}{12}\{f^{″}(y(t_n))(f(y(t_n)){)}^2+(f^{\prime }(y(t_n)){)}^{2}f(y(t_n))\}+O(h^4).$

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.