Consistency improved Euler method

Question

I have the butcher tablaeu for the improved Euler method

\begin{array} {cc|c} 0 & 0 & 0 \\ 1 &0 & 1 \\ \hline \frac{1}{2} &\frac{1}{2} \end{array}

I need to show that this method is consistent and also of order 2.

For consistency , I understand I need to show that the truncation error tends towards $0$ however I’m unsure how to go about this.

Answer 1

The method is given by $$ y_{n+1} = y_n + \frac h2 (f(t_n,y_n) + f(t_n+h,y_n + h f(t_n,y_n)) $$ so the error at $t_{n+1} = t_n+h$ is given by $$ y_{n+1}-y(t_{n+1})=y_n+\frac h2 f(t_n,y_n)+\frac h2 f(t_n+h,y_n+hf_n)-y(t_n+h) $$

Now you must work the RHS... Ingredients:

1. Taylor's formula for functions of one variable to work on $y(t_{n+1})$
2. Taylor's formula for functions of two variables to work on $f(t_n+h,y_n+h f_n)$.
3. Chain rule to relate the total (time) derivatives with the partial derivatives of $f$ with respect to $t$ and $y$.
4. Assume that the method is exact at $t_n$, which means that $y_n = y(t_n)$. (local truncation error)