Is everyone ok with the concept of domains of influence and dependence with respect to these pdes? I’m a bit worried that you aren’t.

And shocks/multi-valued solutions.

I'm not OK. Some scary thoughts and flashbacks I had during Newsnight last night were,

1. Upwind FDM smears out sharp fronts/shock, making them smoother

2. Discontinuities get propagated

3. Discontinuous initial conditions

4. Discontinuous initial derivatives

5. Characteristics intersect(is that a shock)

And for some reason "Rankine–Hugoniot conditions" seem to surface now that we had in undergrad fluid dynamics, but it's a long way away. I used conservation of energy (Bernoulli's principle) for waterhammer but I would not know what to do with conservation {mass, momentum), e.g. in a PDE.

But if we stick in bit of the old diffusion, much of 1-5 disappears except (4 and 5?)

How do we know when shocks occur in general?

I have a passing acquaintance with these. My MSc research was FEM for hyperbolic systems of Friedrichs type in Sobolev spaces, but little of 1-5.