I’m reading Finite Difference Schemes and Partial Differential Equations by John Strikwerda (2nd ed., SIAM ,2004).
You can see the first chapter at here. (First chapter is allowed to public for a sample.)
(In this context, $u^1, u^2$ is not the first or second power of $u$. Those are just for distinction. You can see them as $u_1, u_2$. In the following, Subscriptions usually stands for differentiation. So $u_x$ means $\frac{\partial u}{\partial x}$.)
I cannot understand the red box parts.
What is expanding along the characteristics?
And, as 2nd question, How about in Region 5 ? (The textbook’s answer abbreviates it)
The principle behind these two cases is the same. In both cases, the information is located on a boundary of the domain, and we want to compute the solution in the interior of the domain (hence "extending to the interior"). For the first case, the picture is the following:
The red mark has coordinates $(t,x)$ in the $t$-$x$ plane, and the blue mark has coordinates $(t_0,1)$. Both marks are connected through a 2-characteristic curve. Remind that the value of $w^2$ is constant along 2-characteristics. Thus, $$ w^2(t,x) = w^2(t_0,1) \, . $$ Since the information propagates at the speed $-1$ on this curve, one has the relationship $$ \frac{x-1}{t-t_0} = -1 \, , $$ which ends the proof of the first red box part. The second red box is obtained similarly.