Mass Is Conserved Exactly, Yet the Density Is Negative — The Positivity-Preserving Flux Limiter
·9 min read
At CFL 0.45 a limiter-off double rarefaction dies at step 2: density 0.37, pressure -0.163. One θ per face blends LF and the high-order flux.
Supersonic Flow Can't See Upstream — Characteristics and Sound Waves in the Euler Equations
·6 min read
At 220 m/s in air the speeds are -118.06, 220, and 558.06 m/s. Past Mach 1 the left-going sound wave no longer reaches upstream.
What You Give Up the Exact Riemann Solution For — the Roe Solver and the Entropy Fix
·7 min read
When an eigenvalue crosses zero, Roe locks an expansion shock. The sqrt(rho) average and Harten's floor decide whether that sonic kink stays.
Catching Shocks Without Multiplying a Matrix at Every Face — AUSM Flux Splitting
·6 min read
The Shu–Osher left state in the code, density 3.857143 and pressure 10.33333, is the Mach 3 Rankine–Hugoniot state at gamma 1.4.
The Euler Eigensystem in an Arbitrary Direction — Three Waves, Two Repeats, One Singularity
·6 min read
At n_x=0.01 the R-1 left-eigenvector row divides by 100, and at zero it diverges. A 30 degree face keeps R-1; a face along y switches to R-2.