Shock waves!

Where u_s is the shock wave speed, ρ₁ and ρ₂ are the mass density of the fluid behind and inside the shock, u₂ is the particle velocity of the fluid inside the shock, p₁ and p₂ are the pressures in the two regions, and E₁ and E₂ are the internal energies per unit mass in the two regions. Wiki

I am looking at these because they have the dreaded time and distance fake variables.  These should be handled by the recursive power series and eliminated.  But it has all the conservations of stuff and these can generally be solved with the tanh finite differences method, naturally. I am just looking, dunno if I will actaully do much work on this.  But the results should obey the circular state system and thus compute Pi over the total system of state.