RDCON - BUGFIX - Galerkin F built from the factored spline used by the asymptotics

[StuartBenjamin]

RDCON's Galerkin solve and the asymptotic series at each rational surface used two different splines of F. gal_get_fkg built F from fmats_gal, a spline of F. sing_mmat/sing1_mmat use fmats, a spline of its Cholesky factor L (F = L Lᴴ). Between grid points these differ, so the big asymptotic solution does not exactly solve the Galerkin operator. The residual is singular at the surface, and Δ′ picks up an error that jumps with purely numerical settings (nx, dx1/dx2, gal_tol). 3/1 is hit hardest because its Δ′ is a large cancellation.

Fix

gal_get_fkg builds F = (QL)(QL)ᴴ from fmats, with Q = diag(m − nq), so both parts use the same F. fmats_gal is now used only for the bin_fmat diagnostic. Side effect: with delta_mband > 0, the F entries outside the band were left undefined; they are now zero. (K and G outside the band are still unset, a separate older issue.)

Results

D3D-like TokaMaker i-file (mpsi 257, n = 1, sing_order 8, LU), one numerical setting changed at a time:

Δ′(3/1)	develop	this PR
nx 128 / 256 / 512 / 1024	−3.78 / −3.90 / −5.45 / −4.45	−3.812 / −3.811 / −3.815 / −3.830
gal_tol 1e-9 / 1e-11 / 1e-12	−3.88 / −3.72 / −3.83	−3.814 / −3.817 / −3.809
spread over all 10 runs (also dx1/dx2 ×½, ×2 and psihigh + 1e-6)	45 %	0.8 %

Shipped examples, Δ′ range over nx 128–1024:

	develop	this PR
a5_tearing 2/1	13.04 to 13.43	13.16 to 13.22
DIIID_resistive 2/1	3.71 to 4.05	3.81 to 3.84
DIIID_resistive 3/1	−9.63 to −10.07	−10.138 to −10.142

Not improved: DIIID 4/1 (−23.3 to −24.1 on develop, −24.0 to −25.2 here) and the edge surfaces. DIIID 5/1 spreads more here (−3767 to −4570, develop −3383 to −3415), and 6/1 is unconverged in both. All runs still have the sing_mmat slot typo that #304 fixes.

Scripts: rdcon_galerkin_spline_scripts.zip (python with numpy, xarray, netCDF4, matplotlib):

PY_DP=python3 ./sweep.sh DEV/bin/rdcon out_dev
PY_DP=python3 ./sweep.sh PR/bin/rdcon out_fix
python3 plot_dp.py out_dev/dp.txt out_fix/dp.txt dp_galerkin.png

run_examples.sh and plot_examples.py do the same for the shipped examples.

No dependencies; merges cleanly with #301.

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🤖 Generated with Claude Code

[matt-pharr]

Nice clean tweak!