Author here. It is indeed specific to this 2D Poisson matrix, although loop-unrolling + substitution is a very general technique in performance engineering. Eventually, whenever I'm done teaching classes, we'll get to specialized sparse matrix solvers for this problem and we'll see they come with their own set of problems (and loop carried dependencies are definitely one of them).

In any case, the idea behind this series on Jacobi (this is like the 4th post I believe) is to help students (and people interested in scientific computing in general) get what actually matters computationally and lift the hood to get some understanding of the interplay between such algorithms and their actual implementation on hardware.