(a) Define the linear stability domain for a numerical method to solve $\mathbf{y}^{\prime}=\mathbf{f}(t, \mathbf{y})$. What is meant by an A-stable method?

(b) A two-stage Runge-Kutta scheme is given by

$$\mathbf{k}_{1}=\mathbf{f}\left(t_{n}, \mathbf{y}_{n}\right), \quad \mathbf{k}_{2}=\mathbf{f}\left(t_{n}+\frac{h}{2}, \mathbf{y}_{n}+\frac{h}{2} \mathbf{k}_{1}\right), \quad \mathbf{y}_{n+1}=\mathbf{y}_{n}+h \mathbf{k}_{2},$$

where $h$ is the step size and $t_{n}=n h$. Show that the order of this scheme is at least two. For this scheme, find the intersection of the linear stability domain with the real axis. Hence show that this method is not A-stable.