Consider the implicit Runge-Kutta method: \begin{equation*} y_{n+1} = y_n + hf\left(t_n + \frac{2}{3}h, \frac{1}{3}(y_n + 2y_{n+1})\right) \end{equation*}
a) Show that the local truncation error is $O(h^2)$.
b) Show that the method is A-stable.
So far this is what I have done:
I know we should use the local truncation formula
\begin{equation*} \tau(x,h)=\frac{y(x+h)-y(x)}{h}- \phi(x,y(x),h) \end{equation*}
After using Taylor series expansion, I arrive at
\begin{equation*} \phi (x, y(x); h) = f(t, \frac{y}{3}) + \frac{2}{3}hf_x(t, \frac{y}{3}) + \frac{2}{3}(y(t)+hy'(\eta_t))f_y(t, \frac{y}{3}) + O(h^2) \end{equation*}
Then, again by Taylor series expansion
\begin{equation*} \tau(x,h)=y'(t)+\frac{h}{2}y''(\xi_t) - (f(t, \frac{y}{3}) + \frac{2}{3}hf_x(t, \frac{y}{3}) + \frac{2}{3}(y(t)+hy'(\eta_t))f_y(t, \frac{y}{3}) + O(h^2)) \end{equation*}
Am I going in the right direction? I don't know what else I can do from here, and I would also like to know if I should use one more term in Taylor's expansions.
The first step in deriving a Runge-Kutta method is to determine values for $ a_1 $, $\alpha_1 $, and $ \beta_1 $ with the property that $ a_1f(t + \alpha_1, y + \beta_1) $ approximates
$$ T^{(2)}(t, y) = f(t, y) + \frac{1}{2}f'(t, y), $$
with error no greater than $O(h^2)$, which is the same as the order of the local truncation error for the Taylor method of order two. Since $$ f'(t, y) = \frac{df}{dt}(t,y) = \frac{\partial f}{\partial t}(t, y) + \frac{\partial f}{\partial y}(t, y) \cdot y'(t) \quad \text{and} \quad y'(t)=f(t, y), $$ we have $$ T^{(2)}(t, y) = f(t, y) + \frac{h}{2} \frac{\partial f}{\partial t}(t, y) + \frac{h}{2} \frac{\partial f}{\partial y}(t, y) \cdot f(t, y). \quad (5.18) $$
Expanding $f(t + \alpha_1, y + \beta_1)$ in its Taylor polynomial of degree one about $(t, y)$ gives $$ a_1f(t + \alpha_1, y + \beta_1) = a_1f(t, y) + a_1\alpha_1 \frac{\partial f}{\partial t}(t, y) + a_1\beta_1 \frac{\partial f}{\partial y}(t, y) + a_1 R_1(t + \alpha_1, y + \beta_1). \quad (5.19) $$
where $$ R_1(t + \alpha_1, y + \beta_1) = \frac{\alpha_1^2}{2} \frac{\partial^2 f}{\partial t^2}(t, \mu) + \alpha_1\beta_1 \frac{\partial^2 f}{\partial t \partial y}(t, \mu) + \frac{\beta_1^2}{2} \frac{\partial^2 f}{\partial y^2}(\mu, y), \quad (5.20) $$
for some $\zeta$ between $t$ and $t + \alpha_1$ and $\mu$ between $y$ and $y + \beta_1$.
Matching the coefficients of $f$ and its derivatives in Eqs. (5.18) and (5.19) gives the three equations $$ f(t,y)\;:\;a_1 = 1; \quad \frac{\partial f}{\partial t}(t,y)\;:\;a_1\alpha_1 = \frac{h}{2}; \quad \frac{\partial f}{\partial y}\;:\;a_1\beta_1 = \frac{h}{2} f(t, y). $$
The parameters $a_1, \alpha_1,$ and $\beta_1$ are therefore $$ a_1 = 1, \quad \alpha_1 = \frac{h}{2}, \quad \text{and} \quad \beta_1 = \frac{h}{2} f(t, y), $$
so $$ T^{(2)}(t, y) = f \left( t + \frac{h}{2}, y + \frac{h}{2} f(t, y) \right) - R_1 \left( t + \frac{h}{2}, y + \frac{h}{2} f(t, y) \right), $$
and from Eq. (5.20), $$ R_1 \left( t + \frac{h}{2}, y + \frac{h}{2} f(t, y) \right) = \frac{h^2}{8} \frac{\partial^2 f}{\partial t^2}(\zeta, \mu) + \frac{h^2}{4}f(t,y) \frac{\partial^2 f}{\partial t \partial y}(\zeta, \mu) + \frac{h^2}{8} (f(t,y))^2\frac{\partial^2 f}{\partial y^2}(\zeta, y). $$
If all the second-order partial derivatives of $f$ are bounded, then $$ R_1 \left( t + \frac{h}{2}, y + \frac{h}{2} f(t, y) \right) $$ is $O(h^2)$. As a consequence:
The order of error for this new method is the same as that of the Taylor method of order two.
The difference-equation method resulting from replacing $T^{(2)}(t, y)$ in Taylor's method of order two by $f(t + (\frac{h}{2}), y + (\frac{h}{2}) f(t, y)$ is a specific Runge-Kutta method known as the Midpoint method.
Midpoint Method:
$$ w_{0} = \alpha, \quad w_{i+1} = w_i + h f\left(t_i + \frac{h}{2}, w_i + \frac{h}{2} f(t_i, w_i)\right) \quad \text{for } i = 0, 1, \ldots, N - 1. $$
Only three parameters are present in $a_1 f(t + \alpha_1, y + \beta_1)$, and all are needed in the match of $T^{(2)}$. So a more complicated form is required to satisfy the conditions for any of the higher-order Taylor methods.
The most appropriate four-parameter form for approximating $$ T^{(3)}(t, y) = f(t, y) + \frac{h}{2} f'(t, y) + \frac{h^2}{6} f''(t, y) $$ is $$ a_1 f(t, y) + a_2 f(t + \alpha_2, y + \delta_2 f(t, y)), $$ and even with this, there is insufficient flexibility to match the term $$ \frac{h^2}{6} \bigg[\frac{\partial f}{\partial y}(t, y)\bigg]^2 f(t, y), $$ resulting from the expansion of $\frac{h^2}{6}f''(t, y)$. Consequently, the best that can be obtained from using (5.21) are methods with $O(h^2)$ local truncation error.