Guide 01

Model & numerical method

A finite, proof-informed manufactured-force experiment. The current completed run uses AMReX / incflo on fixed nested Cartesian grids in the full periodic box.

Current refined backend

Oliver's completed run uses a 64³ base and four fixed refined levels, reaching 1024³-equivalent spacing only in the innermost cube. The established incflo integrator and AMReX-Hydro multilevel projection advance velocity; a centered numerical derivative, advection term and Laplacian of the localized target construct the prescribed force. This force construction is not identical to every term of the incflo discretization, so agreement must be measured, not assumed.

The maximum timestep is 0.00025 and the forcing-phase ceiling is 0.0375. A separate output clock preserves all 280 native velocity/force states from rest through 0.995. See the completion record and backend validation and refinement notes. The target formulas below are shared; the discrete representation and diagnostic list describe the earlier PhiFlow backend unless explicitly stated otherwise.

Analytical target

Let τ = 1 − t. The paper’s leading core contracts anisotropically while its peak velocity grows:

ℓᵣ ∼ τ^1/2
ℓ_z ∼ τ^1/2−h
U ∼ τ^−1/2−h
E_core ∼ τ^1/2−3h, 0 < h < 1/100.

The solver evaluates the paper’s implicit (q, η, X) map at every cell. Fixed profiles in those coordinates create the contracting inner core, annular transition, and a localized azimuthal exterior. The profile derivation records exactly which formulas come from the paper and where a finite numerical continuation is introduced.

Governing equation

∂_tu + (u · ∇)u − νΔu + ∇p = f, ∇ · u = 0.

The manufactured force combines the target's centered numerical time derivative, advection and viscous terms. This makes trajectory tracking testable, but it does not eliminate discretization differences in the coupled AMReX solver. Growth of the residual force is a limitation, not evidence of a reproduced singularity.

Start from rest

Velocity and force are exactly zero through t = 0.55. A compact-flat smooth cutoff activates the target over 0.55 < t < 0.775; the target is fully active afterward. The solver advances the exact-zero interval analytically.

Earlier PhiFlow representation

Cell-centered periodic Cartesian grid and double precision.
Velocity generated as a centered discrete curl, giving divergence-free target fields to roundoff.
Projected midpoint RK2 by default, with CFL- and viscosity-limited substeps.
Exact periodic FFT Helmholtz projection matched to the centered divergence operator for production runs.
Finite multiscale annular pulse families plus a bounded grid-deconvolution correction at vector-potential level.

Earlier PhiFlow diagnostics

Every saved frame records component maxima, energy, enstrophy, helicity, dissipation, force norms, momentum cancellation, divergence, tracking error, Fourier occupancy, BKM accumulation, and cells per similarity scale. Structure diagnostics additionally measure axial-outflow alignment, upper and lower axial flux, measured versus theoretical core aspect ratio, pulse localization, pulse covariance, exterior leakage, and fixed-mesh clearance. Production runs can also preserve the complete three-component 3D force field in forces.npz for volume rendering and spatial audits.

Interpretation boundary. This is not an independent numerical proof and is not yet the paper’s exact all-order construction. The informative interval ends when either similarity scale spans fewer than four cells or the spectral tail audit fails. Overflow, CFL failure, or an under-resolved spike is a numerical failure—not a positive result.

Primary sources

OpenAI, Finite Time Blowup for Navier–Stokes, especially Sections 3–7, 10, and Appendix B.
PhiFlow, the established PDE framework used for time advancement.