InnerLayer (GGJ) - NEW - rotated-ray collocation backend (:ray), robust to |Q| ~ 500

[matt-pharr]

Summary

Adds a third interchangeable solver to the GGJ inner-layer model — GGJModel{:ray} — and makes it the default. It returns the same parity-projected matching data (Δ₁, Δ₂) in the same deltac convention as :galerkin and :shooting, but remains accurate far into the regime where both existing backends fail: validated to Q = 500i, the territory resistivity scans (|Q| ∝ η^(−1/3)) and rotation physics actually visit.

Method (one paragraph)

The layer equations are continued analytically onto the complex ray x = e^{iθ}s with θ = arg(Q)/4, which makes the parabolic-cylinder WKB exponent exactly real (no oscillation along the contour) and moves the coefficient pseudo-resonance x² ≈ −Q²(G+KF) — which sits on the real axis for imaginary Q — off the path. Δ is an analytic invariant of the contour angle within the Stokes sector, which doubles as a built-in error diagnostic (outward θ-consistency check). On [0, s_m] a global Chebyshev spectral-element collocation BVP is solved in plain variables with parity conditions at the origin and (Δ, c₁, c₂) as bordered unknowns of one sparse LU per parity (rank-3 Woodbury correction for the second parity). The far-field condition reuses the existing, validated inps Wasow kernel (InnerAsymptotics.jl) — evaluated at complex x via new methods in RayAsymptotics.jl, construction untouched — applied at the series radius S and transported inward by an L-stable 2-stage Radau IIA march in the quotient modulo the decaying exponential pair. The damped-zone march arithmetic runs in Complex{Double64}: at large S the near-parallel power-pair geometry amplifies the structured backward error of the ill-conditioned implicit solves into Δ-mixing ~1e-4 at |Q| = 500 in Float64 (a floor generic to any inps-based method handed to double-precision linear algebra at large S).

Validation

Developed and validated in a standalone package before porting; highlights:

• Manufactured bordered problem with prescribed Δ*: agreement 3e-14.
• Fortran rmatch pins and :galerkin cross-check at moderate Q: 1e-5–1e-9.
• Physical q=4 benchmark surface pinned at Q = 100i and 500i.
• Campaign scale (Slurm, 102 equilibria × 65-point Q-grids incl. DIII-D and ITER PFPO cases, ~14k solves): zero θ-consistency flags above 1e-3 at any |Q|; worst Δ₁ drift 5.8e-4, worst Δ₂ 3.7e-6. The legacy Galerkin self-disagrees at O(1) for |Q| ≳ 1 on the same points.
• In-repo: test/runtests_innerlayer.jl gains a :ray testset (agreement with :galerkin at the GW2020 Eq. 55 paper point; q4 pins at 500i; θ-invariance). 19/19 pass.

Default backend change

GGJModel() now defaults to :ray. Blast radius checked: there are no bare GGJModel() call sites in the repo — all 17 benchmark uses and the regression harness pass solver=:galerkin explicitly. The backends take different numerical-knob keywords, so any stale galerkin-knob call fails loudly rather than returning a silently wrong Δ.

Dependencies

• Random (stdlib, zero footprint) promoted from test-extras to [deps] — seeds the decaying-pair extraction (provably Δ-neutral; kept fixed for bit-reproducibility).
• DoubleFloats was already a project dependency; only the using line is new.

Files

• src/InnerLayer/GGJ/RayAsymptotics.jl (new) — complex-x evaluation methods for the existing inps kernel.
• src/InnerLayer/GGJ/Ray.jl (new) — ODE system, WKB-graded mesh, decaying-pair extraction, Radau march, bordered collocation solve, refinement, delta_convergence error-bar battery, profile reconstruction, and the solve_inner(::GGJModel{:ray}, …) interface method.
• GGJ.jl / InnerLayer.jl — backend registration, exports, docs.

A complete replication-grade methods write-up (strong form, cell-by-cell matrix assembly, march schedule derivation, finite-precision floor analysis) exists and will accompany the paper in preparation.

🤖 Generated with Claude Code

[matt-pharr]

@logan-nc @d-burg this is something I have been working on with claude for a few weeks. It has a number of advantages over the galerkin GGJ solver. It uses collocation which is also a finite element method, but I learned that the reason that the galerkin solver sometimes freaks out is because there are poles to the GGJ solutions in Q space, and when you get near those poles the solutions become super oscillatory on the real axis. However, since we are solving this in a complex domain, you can show that the GGJ solutions should be entire in the complex plane, so the mercier terms in the large |X| limit are the same polynomial + coefficient even off of the real-axis (which is where the physically-relevant part of the solution lies). So you can get a much cleaner/quicker exponential decay of the non-limiting terms if you integrate on a ray that is rotated off of the real axis. This also gives way to an entirely different numerical stability check, which is to re-integrate for multiple rotation angles and see that you get the same Delta_even and Delta_odd. Overall this method is far superior to the galerkin one in all of my testing so far, and can safely go up to Q ~ 500i with a maximum error in the final Delta of ~1e-4 which is pretty good for this problem. Error stays ~1e-7 for more reasonable Q. This enables scanning to lower resistivities (Q ~ 1/eta^1/3) and higher rotation rates (or to absurd growth rates). It also makes the solve more robust for lower |Q|. I will do some more testing on this branch before I merge but I am opening it for review now.