Shock Capturing & Riemann Solvers

1950 — When von Neumann Deliberately Smeared the Shock: Artificial Viscosity and Odd-Even Decoupling

The von Neumann term xi^2 (du)^2 kills post-shock sawteeth on a Sod tube. Omit the code's factor 0.25 and the same xi makes Q four times larger.

2026.05.13

Godunov's Method — One Tiny Shock Tube Inside Every Cell

The Lax tube returns star pressure 3.40948 and star velocity 0.62556. Left state 3.528 makes the left wave a rarefaction and the right a shock.

2026.05.18

Where Negative Density Leaks In — A Positivity-Preserving Flux Limiter for Blast Waves

A Le Blanc rarefaction can drive density negative. Per-face θ blends the high-order flux back toward Lax–Friedrichs only as far as the floor requires.

2026.05.20

The Euler Eigensystem in an Arbitrary Direction — Three Waves, Two Repeats, One Singularity

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.

2026.06.03

Advecting Without Splitting Directions — The Bell-Colella-Glaz Unsplit Upwind Scheme

Directional splitting twists a diagonal scalar. Bell-Colella-Glaz keeps the rotating disk's peak with a (1-CFL) slope weight plus a transverse correction.

2026.06.15

Catching Shocks Without Multiplying a Matrix at Every Face — AUSM Flux Splitting

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.

2026.06.19

[Paper Review] Fooling a Nonlinear Flux Into Behaving Linearly — Jin–Xin & Suliciu Relaxation

Thomann (2019) review: linearizing the Riemann problem with a relaxation system and buying stability with the sub-characteristic condition

2026.07.04

What You Give Up the Exact Riemann Solution For — the Roe Solver and the Entropy Fix

When an eigenvalue crosses zero, Roe locks an expansion shock. The sqrt(rho) average and Harten's floor decide whether that sonic kink stays.

2026.07.13

Supersonic Flow Can't See Upstream — Characteristics and Sound Waves in the Euler Equations

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.

2026.07.24

The Outlet Is Open, Yet the Wave Comes Back — NSCBC and the Price of σ

At σ = 0 a duct reflects 0% and keeps a +0.30 pressure offset; at σ = 0.25 the reflection is 1.9% and the offset drops to 0.0007.

2026.08.04

Mass Is Conserved Exactly, Yet the Density Is Negative — The Positivity-Preserving Flux Limiter

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.

2026.08.05

The Shock Never Left the Starting Line — Where Conservative and Primitive Forms Split

Flux-form Burgers returns shock speed 0.5006. Writing velocity times a gradient returns 0.0000, and an eightfold grid still settles near 0.639.

2026.08.18

The Kinked String Was Fine, the Cut String Overshot by 9% — How the 1747 Wave-Equation Fight Ended Numerically

d'Alembert's traveling waves and Bernoulli's sines agree until the series is cut. A jump keeps a 9% overshoot; a smooth bell is already exact by 32 modes.

2026.08.25

The Same Code Gave 0.79 and 2.00 at Once — What Verification Cases Actually Measure

Sod's density L1 order is 0.79 on a discontinuous tube. The same MUSCL–HLLC code gives 2.001 on a smooth wave, and minmod pulls that down to 1.491.

2026.10.07