Two Philosophies for Triangulating a Point Cloud — Bowyer–Watson and Advancing Front
Twelve points become 17 triangles. A sliced airfoil edge is not restored by the empty circumcircle; a pipe or the front has to take that edge back.
When the Mesh Skews, the Laplacian Leaks — Non-Orthogonal Diffusion Correction
The face-area vector splits into an implicit piece and a deferred correction. Minimum correction hits ratio 1 at 45 degrees and stops contracting.
CFL = 1 takes ten thousand steps, CFL = 100 explodes — Switched Evolution Relaxation
Fixed CFL of 3 needs 588 iterations on steady Burgers. SER with p=1 finishes in 34, then the step is Newton only while a finite CFL cap is still in place.
Giving Each Phase Its Own Equations — The Baer–Nunziato Seven-Equation Model
The printed 324 bar is the 1.2 percent left after two huge stiffened-gas terms cancel. The first relaxation step is what pushes that leftover.
[Paper Review] Not Killing Sound at Low Mach — An AP IMEX Wave Scheme
At ε=0.01 on 64 cells the upwind amplitude falls to 1.980e-7 after 12800 steps, while the implicit AP value stays 1.00. The loss scales as 1/ε.
Entropy Stops the Blow-Up — Karlin's Entropic Lattice Boltzmann
When low viscosity pushes τ toward 1/2, BGK overshoots equilibrium. Entropic LBM stops at the α that leaves discrete H unchanged.
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.
Nine Arrows of Maxwell–Boltzmann — Why LBM's Equilibrium Is a Polynomial
Cutting Maxwell–Boltzmann at Hermite order two builds the D2Q9 polynomial, and that same cut leaves the Mach 0.3 ceiling in place.
The Star That Started Moving — Well-Balanced FVM and Angular Momentum in Polar Coordinates
Mis-discretize 2P/r and a quiet star will leak gas out of its center.
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.
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.
A Heart Valve Without a Body-Fitted Mesh — Peskin's Immersed Boundary Method and the Discrete Delta
Peskin's four-point delta sums to one on the grid. The listing's 24 markers on a radius-8 cylinder sit more than two cells apart, so the wall stays gappy.
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.
When FTCS Trembles, Upwind Goes Numb — First-Order Advection, Artificial Viscosity, and the CFL Trade
At c=0.8 the worst FTCS mode grows by about 1.28 each step. Upwind artificial viscosity drops to zero only when c equals 1.
Mach Architect: Learn Compressible Flow by Designing Nozzles
At γ = 1.4 the choke ratio is 0.5283 and the M = 2 area ratio is about 1.6875. Back pressure between those two parks a normal shock inside the nozzle.
FDM vs FEM vs FVM: The Essential Differences Between Three Discretization Techniques
FDM discretizes the operator, FEM projects onto a basis, and FVM balances face fluxes. That choice fixes conservation, shocks, and unstructured meshes.
Numerical Dungeon: Learn Numerical Methods Through a Game
Three HP, five stages: the quiz runs from the CFL condition through Godunov limiters and Riemann problems to Abgrall's multiphase pressure fix.
Compressible Multiphase Flow CFD: Why It's Hard, Why It Matters
Averaging gamma at a gas-liquid interface does not recover the pressure. Near half gas fraction, the mixture sound speed falls to about one hundredth.