I am following Zenginoglu's paper here on page 6-7. I am confused why my attempts to apply hyperboloidal compactification on the characteristics do not agree with the illustration Zenginoglu provided. I think I may have made a mistake when applying the transformation.
To provide some background: the characteristics of the standard wave equation $$ \partial_t^2u - \partial_x^2u = 0 $$ are $x\pm t=c$ for some constant $c$. By applying the hyperboloidal compactification $$ \begin{split} \rho &= \dfrac{x}{1+x}\\ \tau &= t - h(x) \end{split} $$ where $h(x) = \sqrt{S^2 + x^2}$ is the height function (for consistency and computational purposes, set $S=10$), we transform the characteristics in the compactified $\rho/\tau$ coordinates as follows.
I want to verify this claim computationally. Rewriting the characteristics $x\pm t = c$ with the parametrized function $$ r_\pm(t) = (c\mp t, t) $$ and defining the compactified parametrization of the characteristic as $$ \bar{r}_\pm(t) = (\rho, \tau) = \bigg(\dfrac{x_\pm}{1 + x_\pm}, t - h(x_\pm)\bigg) $$ we get $$ \bar{r}_\pm(t) = \bigg(\dfrac{c\mp t}{1 + c\mp t}, t - \sqrt{S^2 + (c\mp t)^2}\bigg). $$ However when I plot one line for $t\in(-100, 100)$ out I get a big mess, and it's not even in the right quadrant. Did I make a mistake with my transformation?
I don't think your $\rho$ is correct. In the article $x = \rho / \Omega$ (see equation (12)) where $\Omega = \Omega(\rho)$ where the zero set corresponds to spatial infinity. What you set above will have $\Omega = (1-\rho)$ which vanishes at $\rho = 1$, so necessarily you have a different picture from what he has. The correct $\Omega$ to use is given in equation (19) of the paper which takes the form
$$ \Omega = \frac12 \left( 1 - \frac{\rho^2}{S^2} \right) $$
where $S$ is a parameter showing how wide the compactified domain should be. In the shown picture, it was chosen with $S = 10$.