Halving the Grid Spacing Tripled the Error — What an Increment-Based Stopping Rule Hides in Jacobi Iteration
Stop Jacobi when the update drops below a tolerance and the error left behind grows as 1/(1-ρ) on finer grids. A 2D heat-conduction plate compares Jacobi, SOR and optimal ω.
Degree six returned 9 where 15 was required — the three Hermite roots that fixed D2Q9
Where c_s² = 1/3 and the weights 4/9, 1/9, 1/36 actually come from, traced all the way back to the degree of a Gauss–Hermite quadrature rule.
Filling the Third Moment Honestly Made the Error 2.8x Worse — What a Forcing Term Actually Owes
On D2Q9, filling the force third moment raises error from 0.053% to 0.150%. The lattice locks half the tensor, and it arrives only at Burnett order.
Refining the Grid 8x Never Moved 2.4674 — Two Constants Fourier Left Behind in 1807
The convergence objection that got Fourier's 1807 memoir rejected comes back on today's grids as two stubborn constants: 8.9490% and pi^2/4.
At tau = 0.51 I read the viscosity 51x too high — the half step left behind by the discrete LBE
At tau=0.51 the lattice reads nu=0.00333, not 0.17. A second-order expansion drops Delta t/2 from tau: that half step is a factor of 51.
[Paper Review] 0.333 in Tension, 0 in Compression — Where SPH Particles Start Pairing Up
The same SPH code holds under compression and pairs in tension. At h/dx = 1.2 the rate is 0.333; freezing the kernel on the initial lattice drives it to 0.
202x More Scalar Dissipation, 13% Less Temperature — Where a Laminar Flamelet Quenches
Scalar dissipation rose 202-fold while peak temperature fell only 13%, then the burning solution vanished. Python pins that quench limit.
[Paper Review] The Same Point Read 0.544 Apart Along Two Paths — Curl Involutions on a Staggered Grid
A staggered grid holds interface curl near 1e-12 for 400 steps. The collocated Rusanov run on the same shear opens a path gap of 0.544.
The bill for a 1001x larger time step — why a semi-implicit pressure scheme is not L∞ stable
A semi-implicit pressure scheme drops sound speed from the CFL limit and gains a 1001x step at M = 0.001. The discontinuity oscillation does not decay.
Pushing the Frame to 0.2 Cut the Viscosity by 5.97% — The Missing Third Moment in LBM Equilibria
On D2Q9, Σf c³ equals ρu for any distribution you can write. The missing ρu³ breaks Galilean invariance and makes measured viscosity depend on the frame.
49.9% of the Dynamic Coefficients Came Out Negative — the Germano Identity and the Averaging Operator
The Germano identity puts the Smagorinsky coefficient negative at half the points. Where the average is taken decides whether the run stays stable.
Pushing the cell Péclet number to 2 drove the solution to -0.33 — where the energy to minimize disappeared
Advection skews the stiffness matrix, so the energy minimum is gone. At cell Péclet 2, Galerkin falls to -0.33; SUPG buys a bound, not symmetry.
I Dropped the −1/2 and the Viscosity Came Back Six Times Too Large — the Δt/2 Left Behind by LBM's Discretisation
One trapezoidal step and one change of variables leave a Δt/2 in viscosity, velocity, and stress. A scalar ODE and a D2Q9 lattice measure all three.
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.
What Jacobi Cannot Kill in 10,000 Sweeps, a V-Cycle Kills in 10 — The Real Secret of Multigrid
On a 64-cell Poisson problem, twenty V-cycles drive the residual below 1e-10 while Jacobi, at four sweeps a step, stalls near 1e-2.
Implicit Diffusion and the Thomas Algorithm — Buying Stability with One Tridiagonal Solve
At r=0.6 explicit diffusion reaches 1e12 in 60 steps. One Thomas sweep keeps the implicit update finite at that same r, and still finite at r=2.
When Characteristics Collide, a Shock Is Born — Burgers and the (x,t) Plane
A sine of amplitude 0.35 and wavenumber 1 breaks at t* ≈ 0.455 when characteristics cross, and the weak solution cuts that fold with one line.
Quadtree AMR — Letting the Mesh Grow Where It Needs To
Level-7 uniform needs 128 cells; one Gaussian keeps 41 leaves, about 32 percent. A quadtree, slope threshold tau, and a 2:1 rule decide which cells split.
When Lax–Wendroff Shook at a Step — Building a 2nd-Order TVD Advection Scheme with Flux Limiters
At CFL 0.5 and 500 steps, Lax–Wendroff overshoots a unit step by 0.181. minmod, van Leer, and MC stay at zero overshoot on the same pulse.
Choosing a CFD Linear Solver: Comparing Convergence Speeds from Jacobi to BiCGSTAB
On a 60-unknown Poisson system Jacobi is still at residual 2.11e-2 after 2000 sweeps, while CG reaches 1.87e-13 in exactly 60 steps.