Consider the initial value problem for an autonomous differential equation

$$y^{\prime}(t)=f(y(t)), \quad y(0)=y_{0} \text { given }$$

and its approximation on a grid of points $t_{n}=n h, n=0,1,2, \ldots$. Writing $y_{n}=y\left(t_{n}\right)$, it is proposed to use one of two Runge-Kutta schemes defined by

$$y_{n+1}=y_{n}+\frac{1}{2}\left(k_{1}+k_{2}\right)$$

where $k_{1}=h f\left(y_{n}\right)$ and

$$k_{2}= \begin{cases}h f\left(y_{n}+k_{1}\right) & \text { scheme I } \\ h f\left(y_{n}+\frac{1}{2}\left(k_{1}+k_{2}\right)\right) & \text { scheme II }\end{cases}$$

What is the order of each scheme? Determine the $A$-stability of each scheme.