Guide 04
Numerical record
The strongest completed trajectories, what they establish, and the claims they do not support.
Current best completed run: Oliver
The homepage features the completed AMReX / incflo trajectory from exact rest through t = 0.995. All 280 scheduled native states are present and passed the velocity/forcing archive audit. The finite appendix-b-axis-v2-localized profile uses three pulse families. “Best” identifies the completed run selected for presentation; it does not certify the force as spatially resolved.
| Completed native run | Value |
|---|---|
| Base grid / fixed levels | 64³ / 5 |
| Stored / active cells | 1,310,720 / 1,179,648 |
| Finest spacing | 2 / 1024, only inside the ±0.0625 core cube |
| Saved states / final step | 280 / 5,510 |
| Maximum timestep / forcing-phase ceiling | 0.00025 / 0.0375 |
| Endpoint peak speed | 7.415989607 model units |
| Endpoint kinetic energy | 0.036935156 model units |
| Archived-force readback discrepancy | 0 across all 280 native states |
| Elapsed run and completion audit | 9.02 hours |
The completion and media record contains source hashes, the complete output schedule, native checks, slice heights and media checksums. Raw snapshots preserve all three velocity and all three force components in double precision; the web movies sample 256² pixels per plane with the finest containing native cell. This is not a full-resolution 3D web export.
Reading the fixed slices
The homepage movies show a vertical x–z plane at y = 0 and a horizontal x–y plane at z = 0. Both planes remain fixed throughout the run, and the force movie uses the same planes as the velocity movie.
Colors use the site's symmetric-log magnitude scale, with fixed limits shared across both planes and all saved times within each quantity. Every saved state is shown at five frames per second, apart from the initial and final holds; no time interpolation, smoothing or arrow overlays are used.
Earlier uniform-grid media
The previous PhiFlow movies and browser-volume explorers remain archived separately: velocity GIF, forcing GIF, velocity 3D, force 3D, and uniform mesh explorer. They do not show the current refined run. Their original publication record is unchanged.
Historical 224³ complete run
An earlier continuous 224³ trajectory starts from exact rest at t = 0 and reaches t = 0.992. Velocity and force vanish through 0.55, smoothly activate through 0.775, and all 63 requested frames complete.
| Endpoint at t = 0.992 | Value |
|---|---|
| Peak speed, solver / target | 3.30140 / 3.24247 |
| Relative tracking error | 1.1057% |
| Divergence L∞ | 7.11 × 10−14 |
| Peak vorticity | 294.52 |
| Total / core kinetic energy | 0.007728 / 0.000188 |
| Radial / axial scale coverage | 5.21 / 5.41 cells |
| Top-third spectral energy | 0.5464% |
| Manufactured-force L2 norm | 36.76 |
Highest spatial frontier
A 288³ late-window run reaches t = 0.995 with 0.7438% tracking error, divergence 1.03 × 10−13, 5.29 radial cells per scale, 0.4631% top-third spectral energy, and peak vorticity 492.51. Because it initializes from the manufactured target at t = 0.94, it is a concentrated-regime resolution test—not a formation history from rest.
Grid refinement
| Grid | Radial cells | Tracking error | Top-third energy | Peak vorticity | Gate |
|---|---|---|---|---|---|
| 128³ | 3.33 | 3.760% | 3.042% | 122.90 | Fail |
| 160³ | 4.16 | 2.122% | 1.379% | 166.43 | Marginal |
| 192³ | 4.99 | 1.431% | 0.714% | 217.78 | Pass |
Across these equal-window runs, endpoint tracking error decreases with observed order 2.39 and top-third spectral energy with order 3.57. The geometry and Fourier audits agree at 192³.
Temporal and projection checks
Five timestep refinements recover order 1.013 for projected Euler and 2.176 for projected midpoint RK2. On a small trajectory, the periodic FFT projection differs from PhiFlow’s sparse conjugate-gradient path by 4.93 × 10−8 in relative L2 while reducing centered-grid divergence to machine precision.
Negative findings that matter
- The manufactured-force norm and its derivatives still grow toward t = 1; the finite surrogate has not reproduced the analytical residual cancellation.
- Force-release controls relax instead of maintaining concentration.
- The theoretical core aspect ratio is only about 1.04 at the canonical endpoint, so lack of a visually needle-like jet is expected.
- Cross-Mac spatial decomposition loses to communication: the fleet is better used for independent trajectories and parameter controls.
- A numerical blow-up, NaN, or CFL collapse is not evidence of a singular solution.
Next comparison
Kay's 128³-base/five-level run uses the same fixed regions, localized profile and output schedule as Oliver. It reaches 2048³-equivalent spacing in the innermost cube. The next spatial comparison must use matched saved times and volume restriction in their overlap. Launch and runtime records →