I am having trouble finding the region of absolute stability for modified Euler method: \begin{align} w^*_{i+1}&=w_i+hf(t_i,w_i) \\ w_{i+1}&=w_i+\frac{h}2\left[f(t_i,w_i)+f(t_{i+1},w^*_{i+1})\right]. \end{align}
DEFINITION: We define a region R of absolute stability for a one-step method as the region in the complex plane satisfying: $$ R = \{\,hκ ∈ C \;:\; |Q(hκ)| < 1\,\} . $$ I don't fully understand the above definition of region of absolute stability and how to apply it. Clear and step by step help would be much appreciated. Thank you
The function $Q$, which more often than not will be a polynomial or rational function, is the factor that approximates the exponential $e^{hκ}$ in the numerical solution of $w'(t)= κw(t)$. In the exact solution $w(t_{i+1})=e^{hκ}w(t_i)$, in the numerical solution $w_{i+1}=Q(hκ)w_i$. In your scheme \begin{align} w^*_{i+1}&=w_i+h·(κw_i)=(1+hκ)w_i \\ w_{i+1}&=w_i+\frac h2 ((κw_i)+κ(1+hκ)w_i)=(1+hκ+\tfrac12(hκ)^2)w_i \end{align}
So $Q(z)=1+z+\frac12z$ and the stability region is $R=\{z\in\Bbb C:|Q(z)|\le 1\}$, which includes the interval $[-2,1]$ on the real axis.