If you are worried about initial/boundary conditions you need to know more about characteristics and what they mean. Then you will know what sort of conditions make sense, which don’t, where you will find solutions, where you won’t, where there will be a conflict, etc.
It’s such an interesting topic that I think you should get a basic book on the subject. The way you are going about this is too random. You are going to miss all the best bits and, more worryingly, get basic things wrong. I’m usually all in favor of reinventing the wheel. But this subject is fun and I don’t want you to miss that!!
And it has a big impact on numerics. These pdes are much harder than the diffusion equation!
You don't say. Most of the famous schemes (Lax-Wendroff, Friedrichs, 1st order upwinding Vol III PW page 1224) don't use characteristics at all for 1st order pde. They are useful for motivation, How do you solve Asian PDEs using characteristics. They are very niche, but do come in handy.
FYI I wrote a simulation system for a nonlinear Waterhammer flow for central heating of the Hague running on a CDC 6600 supercomputer in Fortran.
I think you have missed the point of this thread completely. The real pde is embedded in the Anchor article...and at the end of the day you need numeric boundary conditions.
These pdes are much harder than the diffusion equation!
The real challenge is numerical, especially convectIon+diffusion. That's not news.