Why are hyperbolic pdes considered difficult to solve numerically?

Question

I am asking only out of curiosity, I did sufficient fluid mechanics to be familiar with shockwaves and the such, but I'm wondering how it relates to the numerical qualities of hyperbolic pdes in general. Wikipedia said the Advection Equation is also difficult to solve numerically, as it is hyperbolic, though it seems 'nicer' than say the Riemann Equation on casual inspection.

Answer 1

Some hyperbolic PDE's are inherently ill-posed because the physical conditions which the equations are modeling are actually elliptic equations. For example, when a hyperbolic equation is modeling a wave equation in a medium of high losses, say, a medium with a complex index of refraction $n$ such that $\Re{n} < \Im{n}$, the wave equation is actually elliptic and imposing Cauchy boundary conditions results in an ill-posed problem.

Fortunately, there are numerous methods of transforming such wave equations, either using differential methods, or finite methods, so that such problems result in a stable solution. Such methods are usually designed for a particular application (e.g., wave propagation through optical materials).