Mass
U₁ = ρ
Density is conserved mass per unit volume. Its flux is ρu.
01 · RIEMANN PROBLEM EXPLORER
Two states. One evolving solution.
02 · CONSERVATION FORM
Three quantities cross every cell face.
Momentum
U₂ = ρu
Momentum flux combines advection and pressure: ρu² + p.
Total energy
E = p/(γ−1) + ½ρu²
Energy flux is u(E+p): internal and kinetic energy plus pressure work.
03 · RUSANOV METHOD
Robust shock capturing through controlled diffusion.
At every interface, average the physical flux from both sides, then subtract a dissipative term proportional to the state jump and maximum local wave speed.
F̂ = ½(FL + FR) − ½λmax(UR − UL)
The CFL condition chooses Δt = CFL Δx / max(|u|+a). Each cell is updated from its outgoing and incoming interface fluxes.