Phase 2 (technical), From a uniform-grid wall to a dynamic-rescaling solver

The numerics upgrade for Route A: decision, derivation, and the Spike-0 reconnaissance

Technical companion to NEGATIVE_RESULT_TWO_CURRENCIES.md (which concluded the uniform-grid fitness search) and the planning records ../PHASE2_NUMERICS_PLAN.md, ../PHASE2_SPIKE0_NOTES.md. Status: this records the decision, derivation, and reconnaissance. The solver has since been built and validated: see the results companion TECHNICAL_SPIKE0_RESCALING.md. References in §6.

1. Why a numerics upgrade, and why this one

Phase 1 concluded that on a uniform grid no scalar fitness read off the trusted, pre-artifact window can isolate genuine singular structure: the near-singularity forms below grid scale, so the fitness re-expresses through whatever is resolved (amplitude, buoyancy split, formation time). Two independent currencies failed a pre-committed, cheat-audited gate through the same degeneracy. The honest exit is not a third currency but a solver that resolves the singular region, so a fitness measures real structure.

The field offers three numerical paradigms for exactly our models (Hou–Luo 2D Boussinesq / 3D axisymmetric Euler / the 1D CLM–De Gregorio family):

• (P1) Dynamic (self-similar) rescaling. Integrate the PDE in rescaled variables so a finite-time self-similar blow-up becomes a steady profile held resolved on a fixed grid as physical time approaches T. Classical, originating in the NLS focusing-singularity work of McLaughlin–Papanicolaou–Sulem–Sulem [MPSS86]; it is how Chen–Hou constructed the approximate self-similar profile underpinning their computer-assisted proof [CH22b].
• (P2) Direct profile construction. Solve the self-similar profile equation itself (Newton, or a physics-informed neural network). No time-integration, no resolution wall. This is the 2023–2026 frontier: the first smooth 2D-Boussinesq / 3D-Euler profiles [WLGB23], now pushed to unstable and singular profiles at machine precision [2511.22819].
• (P3) Adaptive mesh refinement. The original Luo–Hou 2014 discovery tool [LH14]. General but heavy; strictly more work than P1 when the blow-up is self-similar.

Decision (with the user, via a reviewed options menu): build P1, reject P3. P1 reuses the validated pseudo-spectral solver, dissolves the below-grid-scale wall, and its late-time state is a self-similar profile, the de-risked on-ramp to P2. Honest caveat, surfaced by the research and recorded for the record: the genuine novelty frontier is P2 profile construction of unstable profiles, not evolutionary search over initial conditions; P1 mainly finds the stable blow-up, which for 2D Boussinesq Chen–Hou already proved [CH22a]. So P1 is framed as the shared substrate, with the "evolve-ICs vs hunt-profiles" question re-decided at a gate after the solver works.

Terminology. This is a Route-A numerics upgrade. It is not roadmap "Route D" (the later Tier-3 computer-assisted-proof leg). AMR-or-rescaling ≠ proof.

2. Spike-first, on a known answer

Per the project's build discipline (validate a new method on a substrate with a known answer before porting it) Spike 0 implements dynamic rescaling in 1D on the gCLM solver (solver/gclm.py), where the self-similar blow-up is explicit, before Spike 1 ports it to 2D Boussinesq.

Model. gCLM: ω_t + a u ω_x = ω u_x, u_x = H(ω), a=0 is CLM [CLM85], a=1 De Gregorio [DG90]. CLM has the closed-form solution ω(x,t) = 4ω₀/((2 − t H ω₀)² + t² ω₀²), blowing up at T* = 2 / max{Hω₀ : ω₀=0}.

Known-answer target (CLM, a=0). Data ω₀ = −sin x → blow-up at x=0, T*=2, and near the singularity ω(x,t) ≈ (T−t)^{−1} Φ(x/(T−t)) with the explicit profile

Φ(η) = −4η / (1 + 4η²).

3. The rescaled formulation (derived, then confirmed against the literature)

Write ω(x,t) = c_ω(τ) W(y,τ), y = x/c_l(τ), dτ/dt = c_ω. The Hilbert transform is scale-invariant (H commutes with dilation), so the rescaled gCLM equation is

W_τ = −β W + δ y W_y − a Ũ W_y + W·HW,     Ũ_y = HW,
β = d log c_ω/dτ,   δ = d log c_l/dτ.

For CLM (a=0) the advection term drops: W_τ = −β W + δ y W_y + W·HW. A self-similar profile is a steady state. In the physical scaling c_ω ~ 1/(T−t), c_l ~ (T−t), i.e. β=1, δ=−1.

This hand derivation was subsequently confirmed identical to the published gCLM dynamic-rescaling scheme of Huang–Tong–Wang [HTW26] (convention ω = C_ω^{−1}Ω, X=C_l x):

Ω_τ + (c_l X + a U) Ω_X = (c_ω + U_X) Ω,     U_X = H(Ω),
U(X,τ) = (1/π) ∫_ℝ ln|(X−Y)/Y| Ω(Y,τ) dY.

with the normalization conditions (non-degenerate, a=0)

Condition 1 (fix slope Ω_X(0)):  c_l = c_ω + (1−a) U_X(0,τ)
Condition 2 (a=0):               c_ω = 1 − U_X(0,τ) = 1 − HΩ(0,τ)
  ⟹ for a=0:  c_l ≡ 1,   c_ω = 1 − HΩ(0),   and at the profile HΩ̄₀(0)=2 ⇒ c_ω → −1.

Exact CLM target in this convention: Ω̄₀(X) = −4X/(1+4X²), HΩ̄₀(X) = 2/(1+4X²). The pair Ω̄₀ ↔ 2/(1+4X²) is a ready-made unit test for the line Hilbert transform.

4. Reconnaissance: three false starts, each a finding

Cheap probes (../../phase2_spike0_probe.py, and a scratch whole-line build) established, before committing to the real solver:

The unifying reading: the true CLM self-similar profile lives on the whole line and decays only like 1/X, and the self-similar dilation X Ω_X has an unbounded advection coefficient. A periodic or uniform spectral discretization cannot carry either. This is exactly why the published scheme uses a stretched, non-uniform grid and a non-FFT Hilbert transform.

5. The build recipe (from [HTW26] Appendix C), and the remaining work

C.1: line Hilbert transform on a non-uniform grid (the crux). Approximate f on [−M,M] with C¹₀ cubic-spline / Hermite basis functions P_i, Q_i chosen so their Hilbert transforms are bounded and closed-form: H(P_i)=A(l)−A(r), H(Q_i)=(x_{i-1}−x_i)B(l)−(x_{i+1}−x_i)B(r), with A(s),B(s) rational-plus-ln|1−s| and Mathematica minimax series near s=0 to avoid cancellation; diagonal terms H(P_i)(x_i)=(1/π)ln|(x_{i-1}−x_i)/(x_{i+1}−x_i)|, H(Q_i)(x_i)=(x_{i-1}−x_{i+1})/(3π). The velocity U is the analytic integral of the same elements. This computes the line H on an arbitrary grid: where FFT cannot go.

Grid: stretched cosh/sinh. X(ρ) = X_m(1−cosh ρ) + √(c+X_m²) sinh ρ, uniform ρ, Δρ=0.01, truncated at M=10¹⁰, X_m=argmax|Ω|, c set so N_bulk=600 points fall in the half-max peak. Key property: X ~ sinh ρ ⇒ dX ~ X·Δρ, so the dilation CFL is dt < dX/X ~ Δρ, independent of M, this is the cure for the uniform-grid death.

C.2: time-stepping. Evolve f = Ω/Xᵏ (k≥3 odd for degenerate profiles; k=1 for the non-degenerate CLM whose profile vanishes to order 1) to protect the origin vanishing; advection via 5th-order WENO [JS96]; time via SSPRK(10,4) [GKS11]; converge at ‖f_τ‖_∞ < 10⁻⁸.

Remaining work (a genuine multi-day solver build). In dependency order: 1. solver/line_hilbert.py + test_line_hilbert.py: the C.1 spline-analytic transform, validated against −4X/(1+4X²) ↔ 2/(1+4X²). The crux; self-contained; hard pass/fail. 2. the stretched grid + WENO/SSPRK time integration of the rescaled equation. 3. test_gclm_rescaled.py, the pre-committed CLM success predicate: Ω → Ω̄₀, c_ω → −1, reconstructed T* → 2, all resolution-stable.

For a POC (Spike 0, ~1% not machine precision) the stretched mesh can be fixed (no AMR) and M ~ 10²–10³; the C.1 transform is essential regardless.

6. Honest scope

Even a flawless Spike-1 solver reproducing the Chen–Hou 2D-Boussinesq profile reproduces a proven, published result in a toy model across "Wall C" (2D Boussinesq ≠ 3D Navier–Stokes; Chen–Hou already proved 2D-Boussinesq boundary blow-up [CH22a]). Reproduction validates our machinery; it is not novel and not a proof. Any genuine novelty is downstream (a new / unstable profile, or one an IC-search surfaces that direct construction had not) and would still require Route D to certify. Nothing here is a blow-up, a proof, or a statement about the Millennium problem. The value is a structure-resolving solver (the enabler that lives on the far side of the uniform-grid wall) and, so far, an honestly-documented reconnaissance of what building it takes.

7. References

• [CLM85] P. Constantin, P. D. Lax, A. Majda, A simple one-dimensional model for the three-dimensional vorticity equation, Comm. Pure Appl. Math. 38 (1985).
• [DG90] S. De Gregorio, On a one-dimensional model for the three-dimensional vorticity equation, J. Stat. Phys. 59 (1990).
• [MPSS86] D. McLaughlin, G. Papanicolaou, C. Sulem, P. Sulem, Focusing singularity of the cubic Schrödinger equation, Phys. Rev. A 34 (1986), dynamic-rescaling origin.
• [LH14] G. Luo, T. Y. Hou, Potentially singular solutions of the 3D axisymmetric Euler equations, PNAS 111 (2014), adaptive-moving-mesh discovery.
• [E21] T. Elgindi, Finite-time singularity formation for C^{1,α} solutions to the incompressible Euler equations on ℝ³, Ann. of Math. 194 (2021).
• [CH22a] J. Chen, T. Y. Hou, Finite time blowup of 2D Boussinesq and 3D Euler equations with C^{1,α} velocity and boundary, arXiv:1910.00173.
• [CH22b] J. Chen, T. Y. Hou, Stable nearly self-similar blowup of the 2D Boussinesq and 3D Euler equations with smooth data, Part I arXiv:2210.07191; Part II (Rigorous Numerics) arXiv:2305.05660.
• [WLGB23] Y. Wang, C.-Y. Lai, J. Gómez-Serrano, T. Buckmaster, Asymptotic self-similar blow-up profile for 3D axisymmetric Euler equations using neural networks, Phys. Rev. Lett. 130, 244002 (2023); arXiv:2201.06780.
• [2511.22819] Resolving Sharp Gradients of Unstable Singularities to Machine Precision via Neural Networks (2025).
• [HTW26] D. Huang, J. Tong, X. Wang, Self-similar finite-time blowups with singular profiles of the generalized Constantin–Lax–Majda model: theoretical and numerical investigations, arXiv:2603.25104, the exact scheme + Appendix C discretization used here.
• [2308.01528] Exact self-similar finite-time blowup of the Hou–Luo model with smooth profiles (2023).
• [JS96] G.-S. Jiang, C.-W. Shu, Efficient implementation of weighted ENO schemes, J. Comput. Phys. 126 (1996): WENO5 (cited as [JW99] in [HTW26]).
• [GKS11] S. Gottlieb, D. Ketcheson, C.-W. Shu, Strong Stability Preserving Runge–Kutta and Multistep Time Discretizations, World Scientific (2011), SSPRK(10,4).